bluuebunny/crossref_metadata_2025_split huggingface dataset:
A remark on normal derivations
Authors: Duggal B. 
Published: July 2002

Given a Hilbert space

H
H

, let

A
,
S

A,S

be operators on

H
H

. Anderson has proved that if

A
A

is normal and

A
S
=
S
A

AS=SA

, then

‖
A
X
−
X
A
+
S
‖
≥
‖
S
‖

\|AX-XA+S\|\ge \|S\|

for all operators

X
X

. Using this inequality, Du Hong-Ke has recently shown that if (instead)

A
S
A
=
S

ASA=S

, then

‖
A
X
A
−
X
+
S
‖
≥
‖
A

‖

−
2

‖
S
‖

\|AXA-X+S\|\ge \|A\|^{-2}\|S\|

for all operators

X
X

. In this note we improve the Du Hong-Ke inequality to

‖
A
X
A
−
X
+
S
‖
≥
‖
S
‖

\|AXA-X+S\|\ge \|S\|

for all operators

X
X

. Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.
The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces
Authors: Howard Ralph 
Published: July 2002

Let

(
M
,
g
)

(M,g)

be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature

K
≤
−
1

K\le -1

. If

f
f

is a compactly supported function of bounded variation on

M
M

, then

f
f

satisfies the Sobolev inequality
\[

4
π

∫
M

f
2

d
A
+

(

∫
M

|

f

|

d
A
)

2

≤

(

∫
M

‖
∇
f
‖

d
A
)

2

.

4\pi \int _M f^2\,dA+ \left (\int _M |f|\,dA \right )^2\le \left (\int _M\|\nabla f\|\,dA \right )^2.

\]
Conversely, letting

f
f

be the characteristic function of a domain

D
⊂
M

D\subset M

recovers the sharp form

4
π
A
(
D
)
+
A
(
D

)
2

≤
L
(
∂
D

)
2

4\pi A(D)+A(D)^2\le L(\partial D)^2

of the isoperimetric inequality for simply connected surfaces with

K
≤
−
1

K\le -1

. Therefore this is the Sobolev inequality “equivalent” to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces. Under the same assumptions on

(
M
,
g
)

(M,g)

, if

c
:
[
a
,
b
]
→
M

c\colon [a,b]\to M

is a closed curve and

w
c

(
x
)

w_c(x)

is the winding number of

c
c

about

x
x

, then the Sobolev inequality implies
\[

4
π

∫
M

w
c
2

d
A
+

(

∫
M

|

w
c

|

d
A
)

2

≤
L
(
c

)
2

,

4\pi \int _M w_c^2\,dA+ \left (\int _M|w_c|\,dA \right )^2\le L(c)^2,

\]
which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature

≤
−
1

\le -1

.
A note on a question of J. Nekovár and the Birch and Swinnerton-Dyer Conjecture
Authors: Ono Ken 
Published: July 2002

If

D
D

is a square-free integer, then let

E
(
D
)

E(D)

denote the elliptic curve over

Q

\mathbb {Q}

given by the equation

({1})

E
(
D
)
:
D

y

2

=
4

x

3

−
27.

\begin{equation*}E(D): Dy^{2}=4x^{3}-27. \tag {{1}} \end{equation*}

Let

L
(
E
(
D
)
,
s
)

L(E(D),s)

denote the Hasse-Weil

L
L

-function of

E
(
D
)

E(D)

, and let

L

∗

(
E
(
D
)
,
1
)

L^{*}(E(D),1)

denote the ‘algebraic part’ of the central critical value

L
(
E
(
D
)
,
1
)

L(E(D),1)

. Using a theorem of Sturm, we verify a congruence conjectured by J. Nekovář. By his work, if

S
(
3
,
E
(
D
)
)

S(3,E(D))

denotes the 3-Selmer group of

E
(
D
)

E(D)

and

D
≠
1

D\neq 1

is a square-free integer with

|

D

|

≡
1

(
mod

3
)

|D|\equiv 1\pmod 3

, then we find that

L

∗

(
E
(
D
)
,
1
)
≢
0

(
mod

3
)
⟺
S
(
3
,
E
(
D
)
)
=
0.

\begin{equation*}L^{*}(E(D),1)\not \equiv 0 \pmod 3 \Longleftrightarrow S(3,E(D))=0. \end{equation*}
Degrees of high-dimensional subvarieties of determinantal varieties
Authors: Sethuraman B. 
Published: July 2002

Let

n
=

p
a

b

n = p^ab

, where

p
p

is a prime, and

g.c.d.
(
p
,
b
)
=
1

\text {g.c.d. }(p,b)=1

. In

P

n
2

−
1

\mathbf {P}^{n^2-1}

, let

X
r

X_r

be the variety defined by

rank

(
(

x

i
,
j

)
)
≤
n
−
r

\text {rank}\, ((x_{i,j})) \le n-r

. We show that any subvariety of

X
r

X_r

of codimension less than

p
a

r

p^ar

must have degree a multiple of

p
p

. We also show that the bounds on the codimension in our results are strict by exhibiting subvarieties of the appropriate codimension whose degrees are prime to

p
p

.
On the Poincaré series and cardinalities of finite reflection groups
Authors: Stembridge John 
Published: July 2002

Let

W
W

be a crystallographic reflection group with length function

ℓ
(
⋅
)

\ell (\cdot )

. We give a short and elementary derivation of the identity

∑

w
∈
W

q

ℓ
(
w
)

=
∏
(
1
−

q

ht
⁡
(
α
)
+
1

)

/

(
1
−

q

ht
⁡
(
α
)

)

\sum _{w\in W}q^{\ell (w)}=\prod (1-q^{\operatorname {ht} (\alpha )+1})/(1-q^{\operatorname {ht}(\alpha )})

, where the product ranges over positive roots

α
\alpha

, and

ht
⁡
(
α
)

\operatorname {ht} (\alpha )

denotes the sum of the coordinates of

α
\alpha

with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit

q
→
1

q\to 1

; i.e.,

|

W

|

=
∏
(
ht
⁡
(
α
)
+
1
)

/

ht
⁡
(
α
)

|W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha )

.
The explicit solution of a diffusion equation with singularity
Authors: Watanabe Michiaki , Watanabe Shuji 
Published: July 2002

We give the explicit solution of the Cauchy problem for the diffusion equation with a singular term:
\[

(
∂

/

∂
t
)

u
=
(
∂

/

∂
x

)
2

u
−
(
k

/

x
2

)

u

,

t
>
0

,

x
∈

R

1

;

(\partial / \partial t ) \, u = ( \partial / \partial x )^2 \, u - ( k / x^2 ) \, u \; , \quad t > 0 \; , \quad x \in \mathbf {R}^1 \; ;

\]

\[

u
(
0
,
x
)
=
f
(
x
)

,

x
∈

R

1

,

u( 0, x) = f(x) \; , \quad x \in \mathbf {R}^1 \; ,

\]
where

k
>
−
1

/

4

k > - 1/4

. We construct the solution on the basis of a generalization of the Fourier transform. We next show that the solution is expressed by an analytic semigroup, and examine smoothness of

x
↦
u
(
t
,
x
)

x \mapsto u(t, x)

and continuity of

x
↦
u
(
t
,
x
)

/

x

β

(
β
>
0
)

x \mapsto u(t, x) / x^{\beta }\left ( \beta > 0 \right )

.
Jacobi matrices with absolutely continuous spectrum
Authors: Janas Jan , Naboko Serguei 
Published: July 2002

Let

J
J

be a Jacobi matrix defined in

l
2

l^2

as

R
e
W

Re W

, where

W
W

is a unilateral weighted shift with nonzero weights

λ
k

\lambda _k

such that

lim
k

λ
k

=
1.

\lim _k \lambda _k = 1.

Define the seqences:

ε
k

:=

λ

k
−
1

λ
k

−
1
,

\varepsilon _k:= \frac {\lambda _{k-1}}{\lambda _k} -1,

δ
k

:=

λ
k

−
1

λ
k

,

η
k

:=
2

δ
k

+

ε
k

.

\delta _k:= \frac {\lambda _k -1}{\lambda _k}, \, \, \eta _k:= 2 \delta _k + \varepsilon _k.

If

ε
k

=
O
(

k

−
α

)
,

η
k

=
O
(

k

−
γ

)
,

2
3

>
α
≤
γ
,

α
+
γ
>
3

/

2

\varepsilon _k = O(k^{-\alpha }) , \, \, \eta _k = O(k^{-\gamma }), \, \, \frac {2}{3}> \alpha \leq \gamma , \, \, \alpha + \gamma > 3/2

and

γ
>
3

/

4

\gamma > 3/4

, then

J
J

has an absolutely continuous spectrum covering

(
−
2
,
2
)

(-2,2)

. Moreover, the asymptotics of the solution

J
u
=
λ
u
,

λ
∈

R

Ju = \lambda u, \, \lambda \in \mathbb {R}

is also given.
Rules and reals
Authors: Goldstern Martin , Kojman Menachem 
Published: July 2002

A “

k
k

-rule" is a sequence

A
→

=
(
(

A
n

,

B
n

)
:
n
>

N

)

\vec A=((A_n,B_n): n>\mathbb N)

of pairwise disjoint sets

B
n

B_n

, each of cardinality

≤
k

\le k

and subsets

A
n

⊆

B
n

A_n\subseteq B_n

. A subset

X
⊆

N

X\subseteq \mathbb N

(a “real”) follows a rule

A
→

\vec A

if for infinitely many

n
∈

N

n\in \mathbb N

,

X
∩

B
n

=

A
n

X\cap B_n=A_n

. Two obvious cardinal invariants arise from this definition: the least number of reals needed to follow all

k
k

-rules,

s

k

\mathfrak {s}_k

, and the least number of

k
k

-rules with no real that follows all of them,

r

k

\mathfrak {r}_k

. Call

A
→

\vec A

a bounded rule if

A
→

\vec A

is a

k
k

-rule for some

k
k

. Let

r

∞

\mathfrak {r}_\infty

be the least cardinality of a set of bounded rules with no real following all rules in the set. We prove the following:

r

∞

≥
max
(
cov
⁡
(

K

)
,
cov
⁡
(

L

)
)

\mathfrak {r}_\infty \ge \max (\operatorname {cov}(\mathbb {K}),\operatorname {cov}(\mathbb {L}))

and

r

=

r

1

≥

r

2

=

r

k

\mathfrak {r}=\mathfrak {r}_1\ge \mathfrak {r}_2=\mathfrak {r}_k

for all

k
≥
2

k\ge 2

. However, in the Laver model,

r

2

>

b

=

r

1

\mathfrak {r}_2>\mathfrak {b}=\mathfrak {r}_1

. An application of

r

∞

\mathfrak {r}_\infty

is in Section 3: we show that below

r

∞

\mathfrak {r}_\infty

one can find proper extensions of dense independent families which preserve a pre-assigned group of automorphisms. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over

ω
\omega

. The consistency of such a family is still open.
Remarks on commuting exponentials in Banach algebras
Authors: Schmoeger Christoph 
Published: July 2002

Suppose that

a
a

and

b
b

are elements of a complex unital Banach algebra such that the spectra of

a
a

and

b
b

are

2
π
i

2\pi i

-congruence-free. E.M.E. Wermuth has shown that then
\[

e
a

e
b

=

e
b

e
a

implies that

a
b
=
b
a
.

e^a e^b = e^b e^a \quad \text {implies that} \quad ab = ba.

\]
In this note we use two elementary facts concerning inner derivations on Banach algebras to give a very short proof of Wermuth’s result.
Characterization based on conditional expectations of adjacent order statistics: A unified approach
Authors: Franco M. , Ruiz J. 
Published: July 2002

In this paper, we show a unified approach to the problem of characterizing general distribution functions based on the conditional expectation between adjacent order statistics,

ξ
(
x
)
=
E
(
h
(

X

r
,
n

)
∣

X

r
+
1
,
n

=
x
)

\xi (x)=E(h(X_{r,n})\mid X_{r+1,n}=x)

or

ξ
¯

(
x
)
=
E
(
h
(

X

r
+
1
,
n

)
∣

X

r
,
n

=
x
)

\overline {\xi }(x)=E(h(X_{r+1,n})\mid X_{r,n}=x)

, where

h
h

is a real, continuous and strictly monotonic function. We have the explicit expression of the distribution function

F
F

from the above order mean function,

ξ
\xi

and

ξ
¯

\overline {\xi }

, and we give necessary and sufficient conditions so that any real function can be an order mean function. Our results generalize the results given for the discrete, absolutely continuous and continuous cases. Further, we show stability theorems for these characterizations.
A Liouville type theorem for the Schrödinger operator
Authors: Melas Antonios 
Published: July 2002

In this paper we prove that the equation

Δ
u
(
x
)
+
h
(
x
)
u
(
x
)
=
0

\Delta u(x)+h(x)u(x)=0

on a complete Riemannian manifold of dimension

n
n

without boundary and with nonnegative Ricci curvature admits no positive solution provided that

h
h

is a

C
2

C^2

function satisfying

lim sup

r
→
∞

r

−
2

inf

x
∈

B
p

(
r
)

h
(
x
)
≥
−

b
n

a
2

\limsup _{r\rightarrow \infty }r^{-2}\inf _{x\in B_p(r)}h(x) \geq -b_na^2

and

Δ
h
(
x
)
≥
−

c
n

a
2

\Delta h(x)\geq -c_na^2

where

0
≤
a
>

sup

x
∈
M

h
(
x
)

0\leq a>\sup _{x\in M}h(x)

, and

b
n

,

c
n

b_n,c_n

are constants depending only on the dimension, thus generalizing similar results in P. Li and S. T. Yau (Acta Math. 156 (1986), 153–201), J. Li (J. Funct. Anal. 100 (1991), 233–256) and E. R. Negrin (J. Funct. Anal. 127 (1995), 198–203) in all of which

h
h

is assumed to be subharmonic. We also give a generalization in case the Ricci curvature of

M
M

is not necessarily positive but its negative part has quadratic decay under the additional assumption that

h
h

is unbounded from above.
On the asymptotic stability in functional differential equations
Authors: Ignatyev A. 
Published: July 2002

Consider a system of functional differential equations

d
x

/

d
t
=
f
(
t
,

x

t

)

dx/dt=f(t,x_{t})

where

f
f

is the vector-valued functional. The classical asymptotic stability result for such a system calls for a positive definite functional

V
(
t
,
φ
)

V(t,\varphi )

and negative definite functional

d
V

/

d
t

{dV}/{dt}

. In applications one can construct a positive definite functional

V
V

, whose derivative is not negative definite but is less than or equal to zero. Exactly for such cases J. Hale created the effective asymptotic stability criterion if the functional

f
f

in functional differential equations is autonomous (

f
f

does not depend on

t
t

), and N. N. Krasovskii created such criterion for the case where the functional

f
f

is periodic in

t
t

. For the general case of the non-autonomous functional

f
f

V. M. Matrosov proved that this criterion is not right even for ordinary differential equations. The goal of this paper is to prove this criterion for the case when

f
f

is almost periodic in

t
t

. This case is a particular case of the class of non-autonomous functionals.
The number of knot group representations is not a Vassiliev invariant
Authors: Eisermann Michael 
Published: July 2002

For a finite group

G
G

and a knot

K
K

in the

3
3

-sphere, let

F
G

(
K
)

F_G(K)

be the number of representations of the knot group into

G
G

. In answer to a question of D. Altschuler we show that

F
G

F_G

is either constant or not of finite type. Moreover,

F
G

F_G

is constant if and only if

G
G

is nilpotent. We prove the following, more general boundedness theorem: If a knot invariant

F
F

is bounded by some function of the braid index, the genus, or the unknotting number, then

F
F

is either constant or not of finite type.
The Noetherian property in some quadratic algebras
Authors: Kramer Xenia 
Published: July 2002

We introduce a new class of noncommutative rings called pseudopolynomial rings and give sufficient conditions for such a ring to be Noetherian. Pseudopolynomial rings are standard finitely presented algebras over a field with some additional restrictions on their defining relations—namely that the polynomials in a Gröbner basis for the ideal of relations must be homogeneous of degree 2—and on the Ufnarovskii graph

Γ
(
A
)

\Gamma (A)

. The class of pseudopolynomial rings properly includes the generalized skew polynomial rings introduced by M. Artin and W. Schelter. We use the graph

Γ
(
A
)

\Gamma (A)

to define a weaker notion of almost commutative, which we call almost commutative on cycles. We show as our main result that a pseudopolynomial ring which is almost commutative on cycles is Noetherian. A counterexample shows that a Noetherian pseudopolynomial ring need not be almost commutative on cycles.
The index of a critical point for densely defined operators of type (𝑆₊)_{𝐿} in Banach spaces
Authors: Kartsatos Athanassios , Skrypnik Igor 
Published: July 2002

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type(S+)L(S_{+})_{L}acting from a real, reflexive and separable Banach spaceXXintoX∗.X^{*}.This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given inlp, p>2,l^{p},~p>2,illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type(S+)).(S_{+})).Applications to nonlinear Dirichlet problems have appeared elsewhere.
Regularity theory and traces of 𝒜-harmonic functions
Authors: Koskela Pekka , Manfredi Juan , Villamor Enrique 
Published: July 2002

In this paper we discuss two different topics concerning

A

\mathcal {A}

- harmonic functions. These are weak solutions of the partial differential equation

div
(

A

(
x
,
∇
u
)
)
=
0
,

\begin{equation*}\text {div}(\mathcal {A}(x,\nabla u))=0,\end{equation*}

where

α
(
x
)

|

ξ

|

p
−
1

≤
⟨

A

(
x
,
ξ
)
,
ξ
⟩
≤
β
(
x
)

|

ξ

|

p
−
1

\alpha (x)|\xi |^{p-1}\le \langle \mathcal {A}(x,\xi ),\xi \rangle \le \beta (x) |\xi |^{p-1}

for some fixed

p
∈
(
1
,
∞
)

p\in (1,\infty )

, the function

β
\beta

is bounded and

α
(
x
)
>
0

\alpha (x)>0

for a.e.

x
x

. First, we present a new approach to the regularity of

A

\mathcal {A}

-harmonic functions for

p
>
n
−
1

p>n-1

. Secondly, we establish results on the existence of nontangential limits for

A

\mathcal {A}

-harmonic functions in the Sobolev space

W

1
,
q

(

B

)

W^{1,q}(\mathbb {B})

, for some

q
>
1

q>1

, where

B

\mathbb {B}

is the unit ball in

R

n

\mathbb {R}^n

. Here

q
q

is allowed to be different from

p
p

.
Radial Solutions to a Dirichlet Problem Involving Critical Exponents when 𝑁=6
Authors: Castro Alfonso , Kurepa Alexandra 
Published: July 2002

In this paper we show that, for each

λ
>
0

\lambda > 0

, the set of radially symmetric solutions to the boundary value problem
\[

−
Δ
u
(
x
)

a
m
p
;
=
λ
u
(
x
)
+
u
(
x
)
|
u
(
x
)
|
,

a
m
p
;

a
m
p
;
x
∈
B
:=
{
x
∈

R
6

:
‖
x
>
1
‖
}
,

u
(
x
)

a
m
p
;
=
0
,

a
m
p
;

a
m
p
;
x
∈
∂
B
,

\begin {aligned} -\Delta u(x) &= \lambda u(x) + u(x)\vert u(x)\vert , && x\in B := \{x\in R^6\colon \|x > 1\| \},\\ u(x) &= 0, && x\in \partial B, \end {aligned}

\]
is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Real connective K-theory and the quaternion group
Authors: Bayen Dilip , Bruner Robert 
Published: July 2002

Let

k
o

ko

be the real connective K-theory spectrum. We compute

k

o
∗

B
G

ko_*BG

and

k

o
∗

B
G

ko^*BG

for groups

G
G

whose Sylow 2-subgroup is quaternion of order 8. Using this we compute the coefficients

t
(
k
o

)
∗
G

t(ko)^G_*

of the

G
G

fixed points of the Tate spectrum

t
(
k
o
)

t(ko)

for

G
=
S

l
2

(
3
)

G = Sl_2(3)

and

G
=

Q
8

G = Q_8

. The results provide a counterexample to the optimistic conjecture of Greenlees and May [ J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, Memoirs AMS 543 (1995)], Conj. 13.4, by showing, in particular, that

t
(
k
o

)
G

t(ko)^G

is not a wedge of Eilenberg-Mac Lane spectra, as occurs for groups of prime order.
Covering the integers by arithmetic sequences. II
Authors: Sun Zhi-Wei 
Published: July 2002

Let

A
=
{

a

s

+

n

s

Z

}

s
=
1

k

A= \{a_{s}+n_{s}\mathbb {Z}\}^{k}_{s=1}

(

n

1

⩽
⋯
⩽

n

k

)

n_{1} \leqslant \cdots \leqslant n_{k})

be a system of arithmetic sequences where

a

1

,
⋯
,

a

k

∈

Z

a_{1}, \cdots ,a_{k}\in \mathbb {Z}

and

n

1

,
⋯
,

n

k

∈

Z

+

n_{1},\cdots ,n_{k}\in \mathbb {Z}^{+}

. For

m
∈

Z

+

m\in \mathbb {Z}^{+}

system

A
A

will be called an (exact)

m
m

-cover of

Z

\mathbb {Z}

if every integer is covered by

A
A

at least (exactly)

m
m

times. In this paper we reveal further connections between the common differences in an (exact)

m
m

-cover of

Z

\mathbb {Z}

and Egyptian fractions. Here are some typical results for those

m
m

-covers

A
A

of

Z

\mathbb {Z}

: (a) For any

m

1

,
⋯
,

m

k

∈

Z

+

m_{1},\cdots ,m_{k}\in \mathbb {Z}^{+}

there are at least

m
m

positive integers in the form

Σ

s
∈
I

m

s

/

n

s

\Sigma _{s\in I} m_{s}/n_{s}

where

I
⊆
{
1
,
⋯
,
k
}

I \subseteq \{1,\cdots ,k\}

. (b) When

n

k
−
l

>

n

k
−
l
+
1

=
⋯
=

n

k

n_{k-l}>n_{k-l+1}= \cdots =n_{k}

(

0
>
l
>
k
)

0>l>k)

, either

l
⩾

n

k

/

n

k
−
l

l \geqslant n_{k}/n_{k-l}

or

Σ

s
=
1

k
−
l

1

/

n

s

⩾
m

\Sigma ^{k-l}_{s=1}1/n_{s} \geqslant m

, and for each positive integer

λ
>

n

k

/

n

k
−
l

\lambda >n_{k}/n_{k-l}

the binomial coefficient

(

l
λ

)

\binom l{ \lambda }

can be written as the sum of some denominators

>
1

>1

of the rationals

Σ

s
∈
I

1

/

n

s

−
λ

/

n

k

,
I
⊆
{
1
,
⋯
,
k
}

\Sigma _{s\in I}1/n_{s}- \lambda /n_{k}, I \subseteq \{1,\cdots ,k\}

if

A
A

forms an exact

m
m

-cover of

Z

\mathbb {Z}

. (c) If

{

a

s

+

n

s

Z

}

s
=
1

s
≠
t

k

\{a_{s}+n_{s}\mathbb {Z}\}^{k}_{\substack {s=1\ s\not =t}}

is not an

m
m

-cover of

Z

\mathbb {Z}

, then

Σ

s
∈
I

1

/

n

s

,
I
⊆
{
1
,
⋯
,
k
}
∖
{
t
}

\Sigma _{s\in I}1/n_{s}, I \subseteq \{1,\cdots ,k\}\setminus \{t\}

have at least

n

t

n_{t}

distinct fractional parts and for each

r
=
0
,
1
,
⋯
,

n

t

−
1

r=0,1,\cdots ,n_{t}-1

there exist

I

1

,

I

2

⊆
{
1
,
⋯
,
k
}
∖
{
t
}

I_{1},I_{2} \subseteq \{1,\cdots ,k\}\setminus \{t\}

such that

r

/

n

t

≡

Σ

s
∈

I

1

1

/

n

s

−

Σ

s
∈

I

2

1

/

n

s

r/n_{t} \equiv \Sigma _{s\in I_{1}}1/n_{s}-\Sigma _{s\in I_{2}}1/n_{s}

(mod 1). If

A
A

forms an exact

m
m

-cover of

Z

\mathbb {Z}

with

m
=
1

m=1

or

n

1

>
⋯
>

n

k
−
l

>

n

k
−
l
+
1

=
⋯
=

n

k

n_{1}> \cdots >n_{k-l}>n_{k-l+1}= \cdots =n_{k}

(

l
>
0

l>0

) then for every

t
=
1
,
⋯
,
k

t=1, \cdots ,k

and

r
=
0
,
1
,
⋯
,

n

t

−
1

r=0,1,\cdots ,n_{t}-1

there is an

I
⊆
{
1
,
⋯
,
k
}

I \subseteq \{1,\cdots ,k\}

such that

Σ

s
∈
I

1

/

n

s

≡
r

/

n

t

\Sigma _{s\in I}1/n_{s} \equiv r/n_{t}

(mod 1).
Krull-Schmidt fails for serial modules
Authors: Facchini Alberto 
Published: July 2002

We answer a question posed by Warfield in 1975: the Krull-Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring

R
R

is a free abelian group.
Shadowing orbits of ordinary differential equations on invariant submanifolds
Authors: Coomes Brian 
Published: July 2002

A finite time shadowing theorem for autonomous ordinary differential equations is presented. Under consideration is the case were there exists a twice continuously differentiable function

g
g

mapping phase space into

R

m

\mathbb {R}^{m}

with the property that for a particular regular value

c
\boldsymbol c

of

g
g

the submanifold

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

is invariant under the flow. The main theorem gives a condition which implies that an approximate solution lying close to

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

is uniformly close to a true solution lying in

g

−
1

(
c
)

g^{-1}(\boldsymbol c)

. Applications of this theorem to computer generated approximate orbits are discussed.
Kaehler structures on 𝐾_{𝐂}/(𝐏,𝐏)
Authors: Chuah Meng-Kiat 
Published: July 2002

Let

K
K

be a compact connected semi-simple Lie group, let

G
=

K

C

G = K_{\mathbf C}

, and let

G
=
K
A
N

G = KAN

be an Iwasawa decomposition. To a given

K
K

-invariant Kaehler structure

ω
\omega

on

G

/

N

G/N

, there corresponds a pre-quantum line bundle

L

{\mathbf L}

on

G

/

N

G/N

. Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections

O

(

L

)

{\mathcal O}({\mathbf L})

as a

K
K

-representation space. We defined a

K
K

-invariant

L
2

L^2

-structure on

O

(

L

)

{\mathcal O}({\mathbf L})

, and let

H
ω

⊂

O

(

L

)

H_\omega \subset {\mathcal O}({\mathbf L})

denote the space of square-integrable holomorphic sections. Then

H
ω

H_\omega

is a unitary

K
K

-representation space, but not all unitary irreducible

K
K

-representations occur as subrepresentations of

H
ω

H_\omega

. This paper serves as a continuation of that work, by generalizing the space considered. Let

B
B

be a Borel subgroup containing

N
N

, with commutator subgroup

(
B
,
B
)
=
N

(B,B)=N

. Instead of working with

G

/

N
=
G

/

(
B
,
B
)

G/N = G/(B,B)

, we consider

G

/

(
P
,
P
)

G/(P,P)

, for all parabolic subgroups

P
P

containing

B
B

. We carry out a similar construction, and recover in

H
ω

H_\omega

the unitary irreducible

K
K

-representations previously missing. As a result, we use these holomorphic sections to construct a model for

K
K

: a unitary

K
K

-representation in which every irreducible

K
K

-representation occurs with multiplicity one.