bluuebunny/crossref_metadata_2025 huggingface dataset:
Symmetric nilpotent matrices with maximal rank and a conjecture of Grothendieck-Koblitz
Authors: Chai Ching-Li
Published: June 2010

All pairs

(
p
,
n
)

(p,n)

such that there exists an

n
×
n

n \times n

symmetric matrix

A
A

with entries in the ring

Z

p

{\mathbb {Z}_p}

of

p
p

-adic integers such that

A
n

=
p
⋅
U

{A^n} = p \cdot U

with

U
U

invertible in

M

n
×
n

(

Z

p

)

{M_{n \times n}}({\mathbb {Z}_p})

are determined. It is shown that such matrices

A
A

can be used to construct examples of deformations of abelian varieties.
Nonregular extreme points in the set of Minkowski additive selections
Authors: Przesławski Krzysztof
Published: June 2010

A function

s
:

K

n

→

R

n

s:{\mathcal {K}^n} \to {\mathbb {R}^n}

, defined on the family

K

n

{\mathcal {K}^n}

of all compact convex and nonempty sets in

R

n

{\mathbb {R}^n}

, is called a Minkowski additive selection, provided

s
(
A
+
B
)
=
s
(
A
)
+
s
(
B
)

s(A + B) = s(A) + s(B)

and

s
(
A
)
∈
A

s(A) \in A

, whenever

A
,

B
∈

K

n

A,\;B \in {\mathcal {K}^n}

. We confirm the conjecture [6] that there exist extremal selections which are not regular (

s
s

is regular if

s

(
A
)

∈
ext
⁡
A
,

A
∈

K

n

s\left ( A \right ) \in \operatorname {ext} A,\;A \in {\mathcal {K}^n}

).
The Euler characteristic of projectively flat manifolds with amenable fundamental groups
Authors: Kim Hyuk , Lee Hyunkoo
Published: June 2010

The Euler characteristic of a closed projectively flat manifold with amenable fundamental group is shown to be nonnegative, and in fact zero if we further assume that the developing map is injective and the fundamental group is infinite.
The stable neighborhood theorem and lengths of closed geodesics
Authors: Basmajian Ara
Published: June 2010

We derive a generalized collar lemma, called the stable neighborhood theorem, for nonsimple closed geodesics. As an application, we show that there is a lower bound for the length of a closed geodesic having crossing number

k
k

on a hyperbolic surface. This lower bound only depends on

k
k

and tends to infinity as

k
k

goes to infinity. Also, we show that the shortest nonsimple closed geodesic on a closed hyperbolic surface has (geometric) crossing number bounded above by a constant which only depends on the genus.
Adding dominating reals with the random algebra
Authors: Judah Haim , Shelah Saharon
Published: June 2010

We show that there are two models

M
⊆
N

M \subseteq N

such that by forcing with

(

Random)

M

{({\text {Random)}}^M}

over

N
N

we add dominating reals. This answers a question of A. Miller.
On the minimality of tame models in the isols
Authors: Barback Joseph
Published: June 2010

Based on the work of Hirschfeld, it is known that there is a close connection between models for the

Π
2
0

\Pi _2^0

fragment of arithmetic and homomorphic images of the semiring of recursive functions. This fragment of arithmetic includes most of the familiar results of classical number theory. There is a realization of this fragment in the isols in systems called tame models. In this paper a new proof is given to the following result of Ellentuck and McLaughlin on the minimality of tame models: If two tame models share an infinite element, then the models are equal.
Potential space estimates for Green potentials in convex domains
Authors: Fromm Stephen J.
Published: June 2010

Weak type

(
1
,
1
)

(1,1)

bounds are demonstrated for the operators
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f right-arrow from bar integral nabla Subscript x Baseline nabla Subscript x Baseline upper G left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d y and f right-arrow from bar integral nabla Subscript x Baseline nabla Subscript y Baseline upper G left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d y comma">

f
↦
∫

∇
x

∇
x

G
(
x
,
y
)
f
(
y
)
d
y

and

f
↦
∫

∇
x

∇
y

G
(
x
,
y
)
f
(
y
)
d
y
,

f \mapsto \int {{\nabla _x}{\nabla _x}G(x,y)f(y)dy} \quad {\text {and}}\quad f \mapsto \int {{\nabla _x}{\nabla _y}G(x,y)f(y)dy,}

\]
where

G
G

is the Green operator for the Dirichlet problem for the Poisson equation on a bounded convex domain in

R

n

{\mathbb {R}^n}

. These results are used to investigate smoothing properties of the Green operator in potential spaces. An application is given to the restriction of the potential space to the boundary of the domain.
Free and locally free arrangements with a given intersection lattice
Authors: Yuzvinsky Sergey
Published: June 2010

In previous papers the author characterized free arrangements of hyperplanes by the vanishing of cohomology of the intersection lattice with coefficients in a certain sheaf of graded modules over a polynomial ring. The main result of this paper is that for a locally free arrangement the degrees of nonzero homogeneous components of the cohomology modules are bounded by a number depending only on the intersection lattice. In particular, the Hilbert coefficients of the module of derivations of a locally free arrangement are combinatorial invariants. Another result of the paper asserts that the set of free arrangements is Zariski open in the set of all arrangements with a given intersection lattice.
Endomorphism near-rings of 𝑝-groups generated by the automorphism and inner automorphism groups
Authors: Peterson Gary L.
Published: June 2010

The purpose of this paper is to investigate the equality of the endomorphism near-rings generated by the automorphism group and inner automorphism group of a nonabelian

p
p

-group

G
G

. If the automorphism group of

G
G

is not a

p
p

-group, we find that these near-rings are different. If the automorphism group of

G
G

is a

p
p

-group, examples are given illustrating that these near-rings can be different and can be the same.
Normality of 𝜎-products
Authors: Chiba Keiko
Published: June 2010

We prove that no

σ
\sigma

-product of an uncountable family of nontrivial spaces is hereditarily normal or hereditarily

ω
1

{\omega _1}

-paracompact. Also, if every finite subproduct of a

σ
\sigma

-product is (hereditarily) countably paracompact and (hereditarily) normal, then the

σ
\sigma

-product is (hereditarily) countably paracompact iff it is (hereditarily) normal.
On the entropy norm spaces and the Hardy space 𝑅𝑒𝐻¹
Authors: Lang W. C.
Published: June 2010

R. Dabrowski introduced certain natural multiplier operators which map from the entropy norm spaces of B. Korenblum into the Hardy space

Re
⁡

H
1

\operatorname {Re} {H^1}

. We show that the images of the entropy norm spaces in

Re
⁡

H
1

\operatorname {Re} {H^1}

do not include all of that space.
Topology of Milnor fibers of minimally elliptic singularities
Authors: McEwan Lee J.
Published: June 2010

Fundamental groups of Milnor fibers of some minimally elliptic singularities are computed via globalizations of smoothings. A conjecture of Looijenga-Wahl is verified for triangle singularities, which states that embedding dimension at most seven implies that any smoothing has simply-connected Milnor fiber.
On subadditive functions
Authors: Matkowski Janusz , Świątkowski Tadeusz
Published: June 2010

The main result says that every one-to-one subadditive function

f
:
(
0
,
∞
)
→
(
0
,
∞
)

f:(0,\infty ) \to (0,\infty )

such that

lim

t
→
0

f
(
t
)
=
0

{\lim _{t \to 0}}f(t) = 0

must be continuous everywhere. A construction of a broad class of discontinuous subadditive bijections of

(
0
,
∞
)

(0,\infty )

which are bounded in every vicinity of

0
0

is given. Moreover, a problem of extension of a subadditive function defined in

(
0
,
∞
)

(0,\infty )

to a subadditive even function in

R

\mathbb {R}

is considered
On the affine surface area
Authors: Schütt Carsten
Published: June 2010

It is shown that at least two expressions that extend the definition of the affine surface area to all convex bodies coincide.
A Gysin sequence for semifree actions of 𝑆³
Authors: Saralegi Martin
Published: June 2010

In this work we shall consider smooth semifree (i.e., free outside the fixed point set) actions of

S

3

{{\mathbf {S}}^3}

on a manifold

M
M

. We exhibit a Gysin sequence relating the cohomology of

M
M

with the intersection cohomology of the orbit space

M

/

S

3

M/{{\mathbf {S}}^3}

. This generalizes the usual Gysin sequence associated with a free action of

S

3

{{\mathbf {S}}^3}

.
Cardinal invariants in remainders and variations of tightness
Authors: Arhangel′skii A. V. , Bella A.
Published: June 2010

In this paper certain invariants related to the tightness are considered. Particular attention is given to the set tightness. Relations between properties of remainders and compactifications of a metrizable space are also investigated.
Duality for representations of a Hecke algebra
Authors: Kato Shin-ichi
Published: June 2010

We describe a duality operator for representations of the Hecke algebra of a Weyl group or of an affine Weyl group in terms of a certain involution on this algebra.
On compact perturbations and compact resolvents of nonlinear 𝑚-accretive operators in Banach spaces
Authors: Kartsatos Athanassios G.
Published: June 2010

Several mapping results are given involving compact perturbations and compact resolvents of accretive and m-accretive operators. A simple and straightforward proof is given to an important special case of a result of Morales who has recently improved and/or extended various results by the author and Hirano. Improved versions of results of Browder and Morales are shown to be possible by studying various homotopies of compact transformations.
A function in the Dirichlet space such that its Fourier series diverges almost everywhere
Authors: De Souza Geraldo Soares , Sampson G.
Published: June 2010

An analytic function

F
F

on the disc belongs to

B
B

if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper F StartAbsoluteValue EndAbsoluteValue Subscript upper B Baseline equals integral Subscript 0 Superscript 1 Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue upper F prime left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue d theta d r greater-than normal infinity">

|

|

F

|

|

B

=

∫
0
1

∫
0

2
π

|

F
′

(
r

e

i
θ

)

|

d
θ

d
r
>
∞

||F|{|_B} = \int _0^1 {\int _0^{2\pi } {|{F’}(r{e^{i\theta }})|d\theta \,dr > \infty } }

. Notice that

B
⫋

H
1

⫋

L
1

B \varsubsetneqq {H^1} \varsubsetneqq {L^1}

, where

H
1

{H^1}

is the Hardy space of all analytic functions

F
F

so that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper F StartAbsoluteValue EndAbsoluteValue Subscript upper H Sub Superscript 1 Baseline equals sup Underscript 0 greater-than r greater-than 1 Endscripts integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue upper F left-parenthesis r e Superscript i theta Baseline right-parenthesis EndAbsoluteValue d theta greater-than normal infinity comma">

|

|

F

|

|

H
1

=

sup

0
>
r
>
1

∫
0

2
π

|

F
(
r

e

i
θ

)

|

d
θ
>
∞
,

||F|{|_{{H^1}}} = \sup \limits _{0 > r > 1} \int _0^{2\pi } {|F(r{e^{i\theta }})|\,d\theta > \infty ,}

\]

L
1

{L^1}

is the Lebesgue space of integrable functions on

[
0
,
2
π
]

[0,2\pi ]

, and the inclusion

H
1

⫋

L
1

{H^1} \varsubsetneqq {L^1}

is taken in the sense of boundary values, that is,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F element-of upper H Superscript 1 Baseline right double arrow limit Underscript r right-arrow 1 Endscripts minus normal fraktur upper R upper F left-parenthesis r e Superscript i theta Baseline right-parenthesis element-of upper L Superscript 1">

F
∈

H
1

⇒

lim

r
→
1

−
ℜ
F
(
r

e

i
θ

)
∈

L
1

F \in {H^1} \Rightarrow {\lim _{r \to 1}} - \Re F(r{e^{i\theta }}) \in {L^1}

. Kolmogorov in 1923 showed that there exists an

f
f

in

L
1

{L^1}

so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an

f
f

in

H
1

{H^1}

with an almost everywhere divergent Fourier series. The purpose of this note is to announce. Theorem 1. There exists an

f
∈
B

f \in B

, whose Fourier series diverges a.e. This problem was mentioned to the first author by Professor Guido Weiss.
Combinatorial dimension of fractional Cartesian products
Authors: Blei Ron C. , Schmerl James H.
Published: June 2010

The combinatorial dimension of a fractional Cartesian product is the optimal value of an associated linear programming problem.
Endomorphism rings of nondegenerate modules
Authors: Zhou Zheng Ping
Published: June 2010

Let

R

M

{}_RM

be a left

R
R

-module whose Morita context is nondegenerate,

S
=

End

(

R

M
)

S = {\text {End}}({}_RM)

, and

N
=
Hom
⁡
(

R

M
,
R
)

N = \operatorname {Hom} ({}_RM,R)

. If

R

M

{}_RM

is also nonsingular, then the main results of Khuri (Proc. Amer. Math. Soc. 96 (1986), 553-559) are the following: (1)

S
S

is left (right) strongly modular if and only if any element of

S
S

which has zero kernel in

R

M
(

N
R

)

{}_RM({N_R})

has essential image in

R

M
(

N
R

)

{}_RM({N_R})

; (2)

S
S

is a left (right) Utumi ring if and only if every submodule

R

U

{}_RU

of

R

M

(

U
R

∗

of

N
R

)

{}_RM\;(U_R^{\ast }\;\;{\text {of}}\,{N_R})

such that

U
⊥

=
0

(

⊥

U

∗

=
0
)

{U^ \bot } = 0\;({}^ \bot {U^{\ast }} = 0)

is essential in

R

M
(

N
R

)

{}_RM({N_R})

. In this paper, we show that the same results hold in any nondegenerate Morita context without

R

M

{}_RM

being nonsingular and that

S
S

is right nonsingular if and only if

N
R

{N_R}

is nonsingular.
The filtration equation in a class of functions decreasing at infinity
Authors: Eidus D. , Kamin S.
Published: June 2010

We deal with the Cauchy and external boundary problems for the nonlinear filtration equation with variable density. For each density we define a class

ϕ
\phi

of initial functions

φ
\varphi

, such that for any

φ
∈
ϕ

\varphi \in \phi

the problem is uniquely solvable in some set of functions decreasing at infinity with respect to space variables.
Weighted inequalities for geometric means
Authors: Opic B. , Gurka P.
Published: June 2010

A characterization of weights

u
,
v

u,v

is given for which the geometric mean operator

T
f
(
x
)
=
exp
⁡
(

1
x

∫
0
x

ln

f
(
t
)

d
t
)

Tf(x) = \exp (\tfrac {1} {x}\int _0^x {\ln \;f(t)\,dt)}

, defined for

f
f

positive a.e. on

(
0
,
∞
)

(0,\infty )

, is bounded from

L
p

(
(
0
,
∞
)
;
v

d
x
)

{L^p}((0,\infty );v\,dx)

to

L
q

(
(
0
,
∞
)
;
u

d
x
)
,
p
∈
(
0
,
∞
)

{L^q}((0,\infty );u\,dx),p \in (0,\infty )

and

q
∈
[
p
,
∞
)

q \in [p,\infty )

.
von Neumann’s inequality for commuting, diagonalizable contractions. II
Authors: Lotto B. A. , Steger T.
Published: June 2010

We construct a triple

T
=
(

T
1

,

T
2

,

T
3

)

T = ({T_1},{T_2},{T_3})

of commuting, diagonalizable contractions on

C

5

{{\mathbf {C}}^5}

and a polynomial

p
p

in three variables for which

|

|

p
(
T
)

|

|

>

|

|

p

|

|

∞

||p(T)|| > ||p|{|_\infty }

, where

|

|

p

|

|

∞

||p|{|_\infty }

denotes the supremum norm of

p
p

over the unit polydisk in

C

3

{{\mathbf {C}}^3}

.
Spectral multipliers on Lie groups of polynomial growth
Authors: Alexopoulos G.
Published: June 2010

Let

L
L

be a left invariant sub-Laplacian on a connected Lie group

G
G

of polynomial volume growth, and let

{

E
λ

,
λ
⩾
0
}

\{ {E_\lambda },\lambda \geqslant 0\}

be the spectral resolution of

L
L

and

m
m

a bounded Borel measurable function on

[
0
,
∞
)

[0,\infty )

. In this article we give a sufficient condition on

m
m

for the operator

m
(
L
)
=

∫
0
∞

m
(
λ
)
d

E
λ

m(L) = \smallint _0^\infty m(\lambda )d{E_\lambda }

to extend to an operator bounded on

L
p

(
G
)
,

1
>
p
>
∞

{L^p}(G),\;1 > p > \infty

, and also from

L
1

(
G
)

{L^1}(G)

to weak-

L
1

(
G
)

{L^1}(G)

.
A remark on Dunford-Pettis property in 𝐿₁(𝜇,𝑋)
Authors: Cilia Raffaella
Published: June 2010

We prove that if

X
X

is an

L
∞

{L_\infty }

space, then

L
1

(
μ
,
X
)

{L_1}(\mu ,X)

has the Dunford-Pettis Property.
On nonatomic Banach lattices and Hardy spaces
Authors: Kalton N. J. , Wojtaszczyk P.
Published: June 2010

We are interested in the question when a Banach space

X
X

with an unconditional basis is isomorphic (as a Banach space) to an order-continuous nonatomic Banach lattice. We show that this is the case if and only if

X
X

is isomorphic as a Banach space with

X
(

ℓ
2

)

X({\ell _2})

. This and results of Bourgain are used to show that spaces

H
1

(

T

n

)

{H_1}({{\mathbf {T}}^n})

are not isomorphic to nonatomic Banach lattices. We also show that tent spaces introduced by Coifman, Meyer, and Stein are isomorphic to

Rad

H
1

\operatorname {Rad} \;{H_1}

.
On a minimal normal dilation problem
Authors: Athavale Ameer
Published: June 2010

By combining the theory of disintegration of measures with certain approximation theorems in several complex variables, we derive some positive results concerning a minimal normal dilation problem for operator tuples whose coordinates are special functions of subnormal operators as well as their adjoints.
Equivariant maps for homology representations
Authors: Dotzel Ronald M.
Published: June 2010

If Y is a homotopy representation of the finite group G of order n and X is a finite G-CW complex such that, for each subgroup H of G,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis upper X Superscript upper H Baseline semicolon double-struck upper Z Subscript n Baseline right-parenthesis equals upper H Subscript asterisk Baseline left-parenthesis upper Y Superscript upper H Baseline semicolon double-struck upper Z Subscript n Baseline right-parenthesis">

H
∗

(

X
H

;

Z

n

)
=

H
∗

(

Y
H

;

Z

n

)

{H_ \ast }({X^H};{\mathbb {Z}_n}) = {H_ \ast }({Y^H};{\mathbb {Z}_n})

then there exists a G-map

f
:
X
→
Y

f:X \to Y

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript asterisk Superscript upper H Baseline colon upper H Subscript asterisk Baseline left-parenthesis upper X Superscript upper H Baseline semicolon double-struck upper Z Subscript n Baseline right-parenthesis right-arrow upper H Subscript asterisk Baseline left-parenthesis upper Y Superscript upper H Baseline semicolon double-struck upper Z Subscript n Baseline right-parenthesis">

f
∗
H

:

H
∗

(

X
H

;

Z

n

)
→

H
∗

(

Y
H

;

Z

n

)

f_ \ast ^H:{H_ \ast }({X^H};{\mathbb {Z}_n}) \to {H_ \ast }({Y^H};{\mathbb {Z}_n})

is an isomorphism for each subgroup H.
Hermitian *-Einstein surfaces
Authors: Grantcharov Geo , Muškarov Oleg
Published: June 2010

We study the problem when a compact Hermitian

∗

{\ast }

-Einstein surface

M
M

is Kählerian and show that it is true if

M
M

is additionally assumed to be either Einstein or anti-self-dual. We also prove that if the

∗

{\ast }

-scalar curvature of

M
M

is positive then

M
M

is a conformally Kähler surface with positive first Chern class.
Multiple canonical decompositions of families of operators and a model of quasinormal families
Authors: Catepillán Ximena , Ptak Marek , Szymański Wacław
Published: June 2010

A general method of canonical decompositions of several operatorvalued functions (operators) is presented. A model of a family of doubly commuting quasinormal operators is constructed.
Some fixed point theorems for composites of acyclic maps
Authors: Park Sehie , Singh S. P. , Watson Bruce
Published: June 2010

We obtain fixed point theorems for a new class of multifunctions containing compact composites of acyclic maps defined on a convex subset of a locally convex Hausdorff topological vector space. Our new results are applied to approximatively compact, convex sets or to Banach spaces with the Oshman property.
Degree bounds for inverses of polynomial automorphisms
Authors: Cheng Charles Ching-an , Wang Stuart Sui Sheng , Yu Jie Tai
Published: June 2010

It is known that if

k
k

is a field and

F

:
k
[

X
1

,
…
,

X
n

]
→
k
[

X
1

,
…
,

X
n

]

{\mathbf {F}}:k[{X_1}, \ldots ,{X_n}] \to k[{X_1}, \ldots ,{X_n}]

is a polynomial automorphism, then

deg
⁡
(

F

−
1

)
⩽

(
deg

F

)

n
−
1

\deg ({{\mathbf {F}}^{ - 1}}) \leqslant {(\deg \,{\mathbf {F}})^{n - 1}}

. We extend this result to the case where

k
k

is a reduced ring. Furthermore, if

k
k

is not a reduced ring, we show that for any integer

n
⩾
1

n \geqslant 1

and any integer

λ
⩾
0

\lambda \geqslant 0

there exists a polynomial automorphism

F

{\mathbf {F}}

such that

deg
⁡
(

F

−
1

)
=
λ
+

(
deg

F

)

n
−
1

\deg ({{\mathbf {F}}^{ - 1}}) = \lambda + {(\deg \,{\mathbf {F}})^{n - 1}}

.
A direct proof of the Hofmann-Mislove theorem
Authors: Keimel Klaus , Paseka Jan
Published: June 2010

The Hofmann-Mislove Theorem has attracted increasing interest. Through this note we intend to make it easily accessible by a simple direct proof.
Spectrum of the products of operators and compact perturbations
Authors: Gong Wei Bang , Han De Guang
Published: June 2010

In this paper we will give a complete characterization of the operator

B
B

which satisfies the spectral condition

σ
(
A
B
)
=
σ
(
B
A
)

\sigma (AB) = \sigma (BA)

(resp.

σ
e

(
A
B
)
=

σ
e

(
B
A
)
)

{\sigma _e}(AB) = {\sigma _e}(BA))

for every

A
A

in

L
(
H
)

L(H)

and also a spectral characterizations of the product of finitely many normal (resp. essentially normal) operators.
Fixed sets of unitary 𝐺-manifolds
Authors: Costenoble Steven R. , Steiner Michael , Waner Stefan
Published: June 2010

Using K-theory characteristic numbers we generalize results of Conner and Floyd on the orders of spheres in free unitary

Z

/

p
s

\mathbb {Z}/{p^s}

-bordism. These calculations imply results on the possible fixed sets in actions of

Z

/

p
s

\mathbb {Z}/{p^s}

on unitary manifolds. This in turn generalizes to abelian p-groups.
The degree of regularity of a quasiconformal mapping
Authors: Koskela Pekka
Published: June 2010

T. Iwaniec has conjectured that the derivative of a locally

α
\alpha

-Hölder continuous quasiconformal mapping of

R

n

{\mathbb {R}^n}

is locally integrable to any power

p
>

n

1
−
α

p > \frac {n}{{1 - \alpha }}

. We disprove this conjecture by producing examples of quasiconformal mappings of the plane that are uniformly Hölder continuous with exponent

1
2

>
α
>
1

\frac {1}{2} > \alpha > 1

but whose derivatives are not locally integrable to the power

1

1
−
α

\frac {1}{{1 - \alpha }}

.
Nodal sets for sums of eigenfunctions on Riemannian manifolds
Authors: Donnelly Harold
Published: June 2010

Quantitative versions of unique continuation are proved for finite sums of eigenfunctions of the Laplacian on compact Riemannian manifolds. The results include a lower bound for the order of vanishing, a growth estimate for the supremum on compact balls, and a gradient bound. For real analytic metrics, an upper bound for the Hausdorff measure of the zero set is derived.
On the global existence of solutions of a reaction-diffusion equation with exponential nonlinearity
Authors: Barabanova Assia
Published: June 2010

We generalize the result of Haraux and Youkana concerning the global existence of nonnegative solutions of a reaction-diffusion equation with exponential nonlinearity. We also show the asymptotic behavior of the global solutions as

t
→
∞

t \to \infty

.
Continuity of derivations on 𝐻*-algebras
Authors: Villena A. R.
Published: June 2010

We prove that the separating subspace for a derivation on a nonassociative

H
∗

{H^ \ast }

-algebra is contained in the annihilator of the algebra. In particular, derivations on nonassociative

H
∗

{H^\ast }

-algebras with zero annihilator are continuous.
Amenable actions and weak containment of certain representations of discrete groups
Authors: Kuhn M. Gabriella
Published: June 2010

We consider a countable discrete group

Γ
\Gamma

acting ergodically on a standard Borel space S with quasi-invariant measure

μ
\mu

. Let

π
\pi

be a unitary representation of

Γ
\Gamma

on

L
2

(
S
,
d
μ
,

H

)

{L^2}(S,d\mu ,\mathcal {H})

"nicely" related with S. We prove that if

Γ
\Gamma

acts amenably on S then

π
\pi

is weakly contained in the regular representation.
ACCP in polynomial rings: a counterexample
Authors: Heinzer William J. , Lantz David C.
Published: June 2010

We describe an example showing that ACCP need not extend from a ring to a polynomial ring over it.
Nielsen numbers of homotopically periodic maps on infrasolvmanifolds
Authors: McCord Christopher K.
Published: June 2010

A well-known lower bound for the number of fixed points of a self-map

f
:
X
→
X

f:X \to X

is the Nielsen number

N
(
f
)

N(f)

. Unfortunately, the Nielsen number is difficult to calculate. The Lefschetz number

L
(
f
)

L(f)

, on the other hand, is readily computable but usually does not estimate the number of fixed points. In this paper, we show that on infrasolvmanifolds (aspherical manifolds whose fundamental group has a normal solvable group of finite index),

N
(
f
)
=
L
(
f
)

N(f) = L(f)

when

f
f

is a homotopically periodic map.
On the depth of the associated graded ring
Authors: Guerrieri Anna
Published: June 2010

Let (R, m) be a Cohen-Macaulay local ring of positive dimension d, let I be an

m
−

m -

primary ideal of R. In this paper we individuate some conditions on I that allow us to determine a lower bound for depth

gr
I

(
R
)

{\text {gr}_I}(R)

. It is proved that if

J
⊆
I

J \subseteq I

is a minimal reduction of I such that

λ
(

I
2

∩
J

/

I
J
)
=
2

\lambda ({I^2} \cap J/IJ) = 2

and

I
n

∩
J
=

I

n
−
1

J

{I^n} \cap J = {I^{n - 1}}J

for all

n
≥
3

n \geq 3

, then depth

gr
I

(
R
)
≥
d
−
2

{\text {gr}_I}(R) \geq d - 2

; let us remark that

λ
\lambda

denotes the length function.
Volume densities with the mean value property for harmonic functions
Authors: Hansen W. , Netuka I.
Published: June 2010

On a bounded domain U in

R

d

{\mathbb {R}^d}

containing the origin, probability measures

μ
\mu

which have a density w with respect to Lebesgue measure and satisfy

h
(
0
)
=
∫
h

d
μ

h(0) = \smallint h\;d\mu

for every bounded harmonic function on U are studied. A domain U is constructed such that

inf
w
(
U
)
=
0

\inf w(U) = 0

for any such measure. (This solves a problem proposed by A. Cornea.) If, however, U has smooth boundary, then

μ
\mu

having a density

w
∈

C

∞

(
U
)

w \in {\mathcal {C}^\infty }(U)

which is bounded away from zero on U can be constructed. On the other hand, for arbitrary U it is always possible to choose a strictly positive

w
∈

C

∞

(
U
)

w \in {\mathcal {C}^\infty }(U)

tending to zero at

∂
U

\partial U

.
Isometries of noncommutative metric spaces
Authors: Park Efton
Published: June 2010

A. Connes has shown that a unital

C
∗

{C^ \ast }

-algebra equipped with an unbounded Fredholm module can be viewed as a "noncommutative" metric space. In this paper, the author defines a notion of an isometry of a noncommutative metric space, and computes several examples.
Sen’s theorem on iteration of power series
Authors: Lubin Jonathan
Published: June 2010

In the group of continuous automorphisms of the field of Laurent series in one variable over a field of characteristic

p
>
0

p > 0

, Sen’s Theorem describes the rapidity of convergence to the identity of the sequence formed by taking successive pth powers of a given element. This paper gives a short proof of Sen’s Theorem, utilizing the methods of p-adic analysis in characteristic zero.
A kernel theorem on the space [𝐻_{𝜇}×𝐴;𝐵]
Authors: Koh E. L. , Li C. K.
Published: June 2010

Recently, we introduced a space

[

H
μ

(
A
)
;
B
]

[{H_\mu }(A);B]

which consists of Banach space-valued distributions for which the Hankel transformation is an automorphism (The Hankel transformation of a Banach space-valued generalized function, Proc. Amer. Math. Soc. 119 (1993), 153-163). One of the cornerstones in distribution theory is the kernel theorem of Schwartz which characterizes continuous bilinear functionals as kernel operators. The object of this paper is to prove a kernel theorem which states that for an arbitrary element of

[

H
μ

×
A
;
B
]

[{H_\mu } \times A;B]

, it can be uniquely represented by an element of

[

H
μ

(
A
)
;
B
]

[{H_\mu }(A);B]

and hence of

[

H
μ

;
[
A
;
B
]
]

[{H_\mu };[A;B]]

. This is motivated by a generalization of Zemanian (Realizability theory for continuous linear systems, Academic Press, New York, 1972) for the product space

D

R
n

×
V

{D_{{R^n}}} \times V

where V is a Fréchet space. His work is based on the facts that the space

D

R
n

{D_{{R^n}}}

is an inductive limit space and the convolution product is well defined in

D

K
j

{D_{{K_j}}}

. This is not possible here since the space

H
μ

(
A
)

{H_\mu }(A)

is not an inductive limit space. Furthermore,

D
(
A
)

D(A)

is not dense in

H
μ

(
A
)

{H_\mu }(A)

. To overcome this, it is necessary to apply some results from our aforementioned paper. We close this paper with some applications to integral transformations by a suitable choice of A.
(𝑊,𝑅)-matroids and thin Schubert-type cells attached to algebraic torus actions
Authors: Hu Yi
Published: June 2010

Given a projective variety acted on by an algebraic torus, we introduce the notion of (W, R)-matroids using the fixed-point set W and the set R of equivalence classes of one-parameter subgroups. The (W, R)-matroids provide close links among the geometry of torus orbits and Schubert-type cells, the theory of momentum polyhedra, and the combinatorial geometries. On the way to establishing the main theme of the paper, we showed that there are only finitely many Bialynicki-Birula decompositions induced by infinitely many one-parameter subgroups.
𝐾-groups of solenoidal algebras. I
Authors: Brenken Berndt
Published: June 2010

Multiplication by x determines an automorphism of the compact dual group of

Λ
g

=

Z

[
x
,

x

−
1

]

/

(
g
)

{\Lambda _g} = \mathbb {Z}[x,{x^{ - 1}}]/(g)

for

g
∈

Z

[
x
]

g \in \mathbb {Z}[x]

. We determine the K-groups of the

C
∗

{C^ \ast }

-algebra associated with this dynamical system if g is irreducible and has degree one or two. Partial results are included if the degree of g is three.
Real analytic submanifolds under unimodular transformations
Authors: Gong Xiang Hong
Published: June 2010

We study the normal form of totally real and real analytic submanifolds in

C

n

{\mathbb {C}^n}

under holomorphic unimodular transformations. We also consider the unimodular normal form of real surfaces in

C

2

{\mathbb {C}^2}

near an elliptic complex tangent with the non-vanishing Bishop invariant or near a non-exceptional hyperbolic complex tangent.
Nonseparability and uniform structures in locally compact groups
Authors: Hansel G. , Troallic J.-P.
Published: June 2010

Let G be a locally compact topological group. We prove that if G is not a SIN-group, then the quotient Banach space

U

L

(
G
)

/

U

(
G
)

{\mathcal {U}_L}(G)/\mathcal {U}(G)

contains an isometric linear copy of

l
∞

{l^\infty }

. To get this result, we first establish an extension theorem for (bilaterally) uniformly continuous functions on G.
On a sequence transformation with integral coefficients for Euler’s constant
Authors: Elsner C.
Published: June 2010

Let

γ
\gamma

denote Euler’s constant, and let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s Subscript n Baseline equals left-parenthesis 1 plus one half plus midline-horizontal-ellipsis plus StartFraction 1 Over n minus 1 EndFraction right-parenthesis minus log n left-parenthesis n greater-than-or-equal-to 2 right-parenthesis period">

s
n

=

(

1
+

1
2

+
⋯
+

1

n
−
1

)

−
log
⁡
n

(
n
≥
2
)
.

{s_n} = \left ( {1 + \frac {1}{2} + \cdots + \frac {1}{{n - 1}}} \right ) - \log n\quad (n \geq 2).

\]
We prove by Ser’s formula for the remainder

γ
−

s
n

\gamma - {s_n}

that for all integers

n
≥
1

n \geq 1

and

τ
≥
2

\tau \geq 2

there are integers

μ

n
,
0
,

μ

n
,
1

,
…
,

μ

n
,
n

{\mu _{n,0,}}{\mu _{n,1}}, \ldots ,{\mu _{n,n}}

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu Subscript n comma 0 Baseline s Subscript tau Baseline plus mu Subscript n comma 1 Baseline s Subscript tau plus 1 Baseline plus midline-horizontal-ellipsis plus mu Subscript n comma n Baseline s Subscript tau plus n Baseline equals gamma plus upper O Subscript tau Baseline left-parenthesis left-parenthesis n left-parenthesis n plus 1 right-parenthesis left-parenthesis n plus 2 right-parenthesis bullet midline-horizontal-ellipsis bullet left-parenthesis n plus tau right-parenthesis right-parenthesis Superscript negative 1 Baseline right-parenthesis comma">

μ

n
,
0

s
τ

+

μ

n
,
1

s

τ
+
1

+
⋯
+

μ

n
,
n

s

τ
+
n

=
γ
+

O
τ

(

(
n
(
n
+
1
)
(
n
+
2
)
∙
⋯
∙
(
n
+
τ
)

)

−
1

)
,

{\mu _{n,0}}{s_\tau } + {\mu _{n,1}}{s_{\tau + 1}} + \cdots + {\mu _{n,n}}{s_{\tau + n}} = \gamma + {O_\tau }({(n(n + 1)(n + 2) \bullet \cdots \bullet (n + \tau ))^{ - 1}}),

\]
where the constant in

O
τ

{O_\tau }

depends only on

τ
\tau

. The coefficients

μ

n
,
k

{\mu _{n,k}}

are explicitly given and are bounded by

2

3
n
+
τ
−
1

{2^{3n + \tau - 1}}

.
Existence results for some initial and boundary value problems without growth restriction
Authors: Frigon Marlène , O’Regan Donal
Published: June 2010

In this paper, using the Schauder Fixed Point Theorem, we establish some existence results or initial and boundary value problems for differential equations withouth growth restriction on the right member.
On the exponent of the ideal class group of 𝑄(√-𝑑)
Authors: Pappalardi Francesco
Published: June 2010

Let

m
(
d
)

m(d)

be the exponent of the ideal class group of

Q

(

−
d

)

{\mathbf {Q}}(\sqrt { - d} )

, we establish the bound

m
(
d
)
≫

log
⁡
d

log
⁡
log
⁡
d

m(d) \gg \frac {{\log d}}{{\log \log d}}

for almost all the discriminants d by using uniform asymptotic formulas on the number of

n
≤
x

n \leq x

for which there exists a prime less than s for which n is a quadratic residue.
The diameter conjecture for quasiconformal maps is true in space
Authors: Heinonen Juha
Published: June 2010

The diameter conjecture for quasiconformal maps is a natural generalization of the Hayman-Wu theorem on level sets of a univalent function. Astala, Fernández, and Rohde recently disproved this conjecture in the plane. Here we show it is true in space.
Measurability of unions of certain dense sets
Authors: Alonso Alberto , Rosenblueth Javier F.
Published: June 2010

In this paper we study measurability properties of sets of the form
\[

E
t

=
{
t
+
m

α
1

+
n

α
2

|

m
,
n
∈

Z

}

(
t
∈

R

)

{E_t} = \{ t + m{\alpha _1} + n{\alpha _2}|m,n \in {\mathbf {Z}}\} \quad (t \in {\mathbf {R}})

\]
where

α
1

,

α
2

{\alpha _1},{\alpha _2}

are given real numbers with

α
1

/

α
2

{\alpha _1}/{\alpha _2}

irrational. Sets such as these have played an important role to establish certain fundamental results in measure theory. However, the question of measurability of unions of these sets seems not to have been solved. In an initial guess, no sets C and T seem apparent for which

0
>
m
A
>
m
T

0 > mA > mT

, where m denotes the Lebesgue measure in R and

A
=

∪

t
∈
C

E
t

∩
T

A = { \cup _{t \in C}}{E_t} \cap T

. In fact, we prove that if T is any Lebesgue measurable subset of R, then no such sets can exist: no matter which C we choose, if A is measurable then mA equals 0 or mT. Moreover, if A is a nonmeasurable set, the same applies to its Lebesgue outer measure. However, if we remove the condition on T of being measurable, we provide an example of (nonmeasurable) sets C and T for which the outer measure of A lies in between 0 and the outer measure of T.
Normal derivations in norm ideals
Authors: Kittaneh Fuad
Published: June 2010

We establish the orthogonality of the range and the kernel of a normal derivation with respect to the unitarily invariant norms associated with norm ideals of operators. Related orthogonality results for certain nonnormal derivations are also given.
Bounded point evaluations and polynomial approximation
Authors: Thomson James E.
Published: June 2010

We consider the set of bounded point evaluations for polynomials with respect to the

L
P

{L^P}

-norm for a measure. We give an example of a measure where the corresponding sets of bounded point evaluations vary with the exponent p. The main ingredient is the remarkable work of K. Seip on interpolating and sampling sequences for weighted Bergman spaces.
𝐷-sets and BG-functors in Kazhdan-Lusztig theory
Authors: Zou Yi Ming
Published: June 2010

By using Deodhar’s combinatorial setting and Bernstein-Gelfand projective functors, this paper provides some necessary and sufficient conditions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Kazhdan-Lusztig polynomials.
Krull dimension of modules and involutive ideals
Authors: Coutinho S. C.
Published: June 2010

In this paper we establish an upper bound for the Krull dimension of a module over a Weyl algebra in terms of a geometrical invariant of its characteristic variety, the involutive dimension. This is followed by some examples which show that this inequality may be strict.
A note on interpolation of semigroups
Authors: Kéyantuo V.
Published: June 2010

In this note, we extend the theory of interpolation of semigroups considered by Arendt (1991) and Arendt-Neubrander-Schlotterbeck (1992) to include the fractional order case. We then apply the results to the Schrödinger and wave equations on

L
p

(
Ω
)

{L^p}(\Omega )

where

Ω
⊂

R

N

\Omega \subset {\mathbb {R}^N}

is open with bounded Lebesgue measure.
Small Mathieu group coverings in characteristic two
Authors: Abhyankar Shreeram S. , Yie Ikkwon
Published: June 2010

Explicit equations are given for unramified coverings of the affine line in characteristic two whose Galois groups are the Mathieu groups of degrees 11 and 12 and the automorphism group of the Mathieu group of degree 12.
A lower bound for the class numbers of abelian algebraic number fields with odd degree
Authors: Le Mao Hua
Published: June 2010

Let

Δ
K

,

h
K

,

R
K

{\Delta _K},{h_K},{R_K}

denote the discriminant, the class number, and the regulator of the Abelian algebraic number field

K
=

Q

(
α
)

K = \mathbb {Q}(\alpha )

with degree d, respectively. In this note we prove that if

d
>
1
,
2
∤
d

d > 1,2\nmid d

, and the defining polynomial of

α
\alpha

has exactly

r
1

{r_1}

real zeros and

r
2

{r_2}

pairs of complex zeros, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h Subscript upper K Baseline greater-than w StartRoot StartAbsoluteValue normal upper Delta Subscript upper K Baseline EndAbsoluteValue EndRoot slash 2 Superscript r 1 Baseline left-parenthesis 2 pi right-parenthesis Superscript r 2 Baseline 33 upper R Subscript upper K Baseline log 4 StartAbsoluteValue normal upper Delta Subscript upper K Baseline EndAbsoluteValue">

h
K

>
w

|

Δ
K

|

/

2

r
1

(
2
π

)

r
2

33

R
K

log
⁡
4

|

Δ
K

|

{h_K} > w\sqrt {|{\Delta _K}|} /{2^{{r_1}}}{(2\pi )^{{r_2}}}33{R_K}\log 4|{\Delta _K}|

, where w is the number of roots of unity in K.
Equivalence of 𝐺-algebras for abelian 𝐺
Authors: Turull Alexandre
Published: June 2010

We describe the equivalence classes of central simple G-algebras over an infinite field F for G a finite abelian group, provided the following holds. For each prime p for which the field F has primitive pth roots, the Sylow p-subgroup of G is either cyclic or elementary abelian.
Free involutions on 𝐸_{4𝑚} lattices
Authors: Jastrzebowski Wojtek
Published: June 2010

We determine all the conjugacy classes of traceless involutions on the

E

4
m

{E_{4m}}

lattices. In particular, we show that for every

m
>
2

m > 2

there exist precisely two nonconjugate involutions which induce free

Z

[

Z

2

]

{\mathbf {Z}}[{{\mathbf {Z}}_2}]

-module structures. By inspecting the parity of the

E

4
m

{E_{4m}}

form twisted by any such involution, we deduce that a closed, simply connected, topological 4-manifold with intersection form

E

4
m

{E_{4m}}

supports a locally linear involution if and only if m is odd and the Kirby-Siebenmann invariant of the manifold is trivial.
𝜆-power integrals on the Cantor type sets
Authors: Fu Shushang
Published: June 2010

We introduce the notions of

λ
\lambda

-power dyadic derivatives and

λ
\lambda

-power dyadic integrals, so that, in particular, the Cantor ternery function is an indefinite integral of its derivative. Furthermore, under certain conditions on the integrands we can give a Riemann-type definition to the

λ
\lambda

-power dyadic integral.
The Hardy-Littlewood theorem on fractional integration for Laguerre series
Authors: Kanjin Yūichi , Sato Enji
Published: June 2010

The Hardy-Littlewood theorem on fractional integration for Fourier series says that if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript sigma Baseline g tilde sigma-summation Underscript n not-equals 0 Endscripts StartAbsoluteValue n EndAbsoluteValue Superscript negative sigma Baseline ModifyingAbove g With caret left-parenthesis n right-parenthesis e Superscript i n t">

I
σ

g
∼

∑

n
≠
0

|

n

|

−
σ

g
^

(
n
)

e

int

{I_\sigma }g \sim \sum \nolimits _{n \ne 0} {|n{|^{ - \sigma }}\hat g} (n){e^{\operatorname {int} }}

, then

I
σ

{I_\sigma }

is bounded from

L
p

{L^p}

to

L
q

{L^q}

, where

1
>
p
>
q
>
∞
,

1
q

=

1
p

−
σ

1 > p > q > \infty ,\frac {1}{q} = \frac {1}{p} - \sigma

. We shall establish an analogue of this theorem for the Laguerre function system
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace left-parenthesis StartFraction n factorial Over normal upper Gamma left-parenthesis n plus alpha plus 1 right-parenthesis EndFraction right-parenthesis Superscript one half Baseline upper L Subscript n Superscript alpha Baseline left-parenthesis x right-parenthesis e Superscript minus StartFraction x Over 2 EndFraction Baseline x Superscript StartFraction alpha Over 2 EndFraction Baseline right-brace Subscript n equals 0 Superscript normal infinity">

{

(

n
!

Γ
(
n
+
α
+
1
)

)

1
2

L
n
α

(
x
)

e

−

x
2

x

α
2

}

n
=
0

∞

\left \{ {{{\left ( {\frac {{n!}}{{\Gamma (n + \alpha + 1)}}} \right )}^{\frac {1}{2}}}L_n^\alpha (x){e^{ - \frac {x}{2}}}{x^{\frac {\alpha }{2}}}} \right \}_{n = 0}^\infty

.
The Gaussian-Wahl map for trigonal curves
Authors: Brawner James N.
Published: June 2010

If a curve C is embedded in projective space by a very ample line bundle L, the Gaussian map

Φ

C
,
L

{\Phi _{C,L}}

is defined as the pull-back of hyperplane sections of the classical Gauss map composed with the Plücker embedding. When

L
=
K

L = K

, the canonical divisor of the curve C, the map is known as the Gaussian-Wahl map for C. We determine the corank of the Gaussian-Wahl map to be

g
+
5

g + 5

for all trigonal curves (i.e., curves which admit a 3-to-1 mapping onto the projective line) by examining the way in which a trigonal curve is naturally embedded in a rational normal scroll.
Conformal diffeomorphisms preserving the Ricci tensor
Authors: Kühnel W. , Rademacher H.-B.
Published: June 2010

We characterize semi-Riemannian manifolds admitting a global conformal transformation such that the difference of the two Ricci tensors is a constant multiple of the metric. Unless the conformal transformation is homothetic, the only possibilities are standard Riemannian spaces of constant sectional curvature and a particular warped product with a Ricci flat Riemannian manifold.
A rim-metrizable continuum
Authors: Nikiel J. , Treybig L. B. , Tuncali H. M.
Published: June 2010

A locally connected rim-metrizable continuum is constructed which admits a continuous mapping onto a non rim-metrizable space.
Hyperplane sections of arithmetically Cohen-Macaulay curves
Authors: Walter Charles H.
Published: June 2010

We show that for every

r
≥
4

r \geq 4

there exists a

d
r

{d_r}

such that for all

d
≥

d
r

d \geq {d_r}

a general set of r points in

P

r
−
1

{{\mathbf {P}}^{r - 1}}

is not a hyperplane section of an arithmetically Cohen-Macaulay local complete intersection curve in

P

r

{{\mathbf {P}}^r}

. Explicit values for the bound

d
r

{d_r}

are given. In particular, for

r
≥
12

r \geq 12

we have

d
r

=
r
+
3

{d_r} = r + 3

, and this bound is exact.
The qualitative analysis of a dynamical system modeling the formation of two-layer scales on pure metals
Authors: Baker R. L.
Published: June 2010

F. Gesmundo and F. Viani have modeled the growth rates of two-oxide scales by the system:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d q 1 Over d t EndFraction equals m StartFraction upper K 1 Over 2 q 1 EndFraction minus StartFraction m minus 1 Over m EndFraction StartFraction upper K 2 Over 2 q 2 EndFraction comma StartFraction d q 2 Over d t EndFraction equals minus m StartFraction upper K 1 Over 2 q 1 EndFraction plus StartFraction upper K 2 Over 2 q 2 EndFraction period">

d

q
1

d
t

=
m

K
1

2

q
1

−

m
−
1

m

K
2

2

q
2

,

d

q
2

d
t

=
−
m

K
1

2

q
1

+

K
2

2

q
2

.

\frac {{d{q_1}}}{{dt}} = m\frac {{{K_1}}}{{2{q_1}}} - \frac {{m - 1}}{m} \frac {{{K_2}}}{{2{q_2}}},\qquad \frac {{d{q_2}}}{{dt}} = - m\frac {{{K_1}}}{{2{q_1}}} + \frac {{{K_2}}}{{2{q_2}}}.

\]
We provide a complete qualitative analysis of (1.1) by making use of known results about the general n-dimensional dynamical system:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d p Subscript i Baseline Over d t EndFraction equals minus sigma-summation Underscript j equals 1 Overscript n Endscripts StartFraction a Subscript i j Baseline Over p Subscript j Baseline EndFraction comma p Subscript i Baseline left-parenthesis t right-parenthesis greater-than 0 comma i equals 1 comma ellipsis comma n period">

d

p
i

d
t

=
−

∑

j
=
1

n

a

i
j

p
j

,

p
i

(
t
)
>
0
,

i
=
1
,
…
,
n
.

\frac {{d{p_i}}}{{dt}} = - \sum \limits _{j = 1}^n {\frac {{{a_{ij}}}}{{{p_j}}},\quad {p_i}(t) > 0,\qquad i = 1, \ldots ,n.}

\]
We show that for

m
>
1

m > 1

, the Gesmundo-Viani system admits a unique parabolic solution

q
i

(
t
)
=

c
i

t

,

c
i

>
0

{q_i}(t) = {c_i}\sqrt t ,{c_i} > 0

. This parabolic solution attracts all other solutions. Every solution extends uniquely to a solution on

[
0
,
+
∞
)

[0, + \infty )

, such that the extended solution is eventually monotonically increasing. Finally, the trajectory of any solution coincides with a trajectory of the following linear system:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction d q 1 Over d t EndFraction equals minus StartFraction m minus 1 Over m EndFraction StartFraction upper K 2 Over 2 EndFraction q 1 plus m StartFraction upper K 2 Over 2 EndFraction q 2 comma StartFraction d q 2 Over d t EndFraction equals StartFraction upper K 2 Over 2 EndFraction q 1 plus m StartFraction upper K 1 Over 2 EndFraction q 2 period">

d

q
1

d
t

=
−

m
−
1

m

K
2

2

q
1

+
m

K
2

2

q
2

,

d

q
2

d
t

=

K
2

2

q
1

+
m

K
1

2

q
2

.

\frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.

\]
Non-Smirnov domains
Authors: Øyma Knut
Published: June 2010

If

Ω
\Omega

is a Jordan domain, a small perturbation of the boundary gives a non-Smirnov domain.
Polynomial harmonic morphisms between Euclidean spheres
Authors: Eells James , Yiu Paul
Published: June 2010

A characterization is given of the harmonic morphisms between euclidean spheres whose component functions are harmonic homogeneous polynomials of the same degree, and also of polynomial harmonic morphisms between euclidean spaces which map spheres into spheres. These turn out to be isometric to the classical Hopf fibrations.