bluuebunny/crossref_metadata_2025 huggingface dataset:
Necessary and sufficient conditions for the solvability of a nonlinear two-point boundary value problem
Authors: Mawhin J. , Ward J. R. , Willem M.
Published: June 2010

The dual least action principle is used to prove a necessary and sufficient condition for the solvability of a Dirichlet problem of the form

u
+
u
+
f

(

x
,
u

)

=
0

u + u + f\left ( {x,u} \right ) = 0

.

u
(
0
)
=
u
(
π
)
=
0

u(0) = u(\pi ) = 0

when

f

(

x
,
⋅

)

f\left ( {x, \cdot } \right )

is nondecreasing and

∫
0
u

f

(

x
,
v

)

d
v

\int _0^u {f\left ( {x,v} \right )dv}

satisfies a suitable growth condition.
Subhomogeneous AF 𝐶*-algebras and their Fubini products
Authors: Kye Seung-Hyeok
Published: June 2010

We give a characterization of subhomogeneous AF

C
∗

{C^ * }

-algebras in terms of their

C
∗

{C^*}

-subalgebras. Also, we show that an AF

C
∗

{C^*}

-algebra is a

C
∗

{C^*}

-algebra with trivial Fubini products if and only if it is subhomogeneous.
Minimal disks and compact hypersurfaces in Euclidean space
Authors: Moore John Douglas , Schulte Thomas
Published: June 2010

Let

M
n

{M^n}

be a smooth connected compact hypersurface in

(
n
+
1
)

(n + 1)

-dimensional Euclidean space

E

n
+
1

{E^{n + 1}}

, let

A

n
+
1

{A^{n + 1}}

be the unbounded component of

E

n
+
1

−

M
n

{E^{n + 1}} - {M^n}

, and let

κ
1

⩽

κ
2

⩽
⋯
⩽

κ
n

{\kappa _1} \leqslant {\kappa _2} \leqslant \cdots \leqslant {\kappa _n}

be the principal curvatures of

M
n

{M^n}

with respect to the unit normal pointing into

A

n
+
1

{A^{n + 1}}

. It is proven that if

κ
2

+
⋯
+

κ
n

>
0

{\kappa _2} + \cdots + {\kappa _n} > 0

, then

A

n
+
1

{A^{n + 1}}

is simply connected.
A cancellation criterion for finite-rank torsion-free abelian groups
Authors: Stelzer J.
Published: June 2010

In this paper, a necessary ring-theoretical criterion is given for a finite-rank torsion-free abelian group to have the cancellation property. This generalizes results obtained by L. Fuchs and F. Loonstra [5] for the rank-one case and resolves the cancellation problem for strongly indecomposable groups.
Inner derivations of division rings and canonical Jordan form of triangular operators
Authors: Đoković Dragomir Ž.
Published: June 2010

Let

D
D

be a division ring and

k
k

its center. We show that a generalized canonical Jordan form exists for triangularizable matrices

A
A

over

D
D

which are algebraic over

k
k

, i.e, satisfy

f
(
A
)
=
0

f(A) = 0

for some nonzero polynomial

f
f

over

k
k

. This canonical form is a direct sum of generalized Jordan blocks

J
m

(
α
,
β
)

{J_m}(\alpha ,\beta )

. This block is an

m
m

by

m
m

matrix whose diagonal entries are equal to

α
\alpha

, those on the first superdiagonal are equal to

β
\beta

, and all other entries are equal to zero. If

α
\alpha

is separable over

k
k

then we can choose

β
=
1

\beta = 1

, but in general this cannot be done.
Arens product and the algebra of double multipliers
Authors: Wong Pak Ken
Published: June 2010

Let

A
A

be a semisimple Banach algebra and

M
(
A
)

M(A)

the algebra of double multipliers of

A
A

. We show that

M
(
A
)

M(A)

is isomorphic to

(

A

∗
∗

,
∘
)

({A^{ * * }}, \circ )

if and only if

A
A

has the following properties: (1)

A
A

is Arens regular, (2)

A
A

has a weak approximate identity, and (3)

π
(
A
)

\pi (A)

is an ideal of

(

A

∗
∗

,
∘
)

({A^{ * * }}, \circ )

.
Approximate innerness of positive linear maps of finite von Neumann algebras
Authors: Takemoto Hideo
Published: June 2010

Let

M
M

be a

σ
\sigma

-finite, finite von Neumann algebra with a faithful, normalized, normal trace Tr and a positive linear map

ρ
\rho

of

M
M

into itself. If

ρ
\rho

is approximately inner with respect to the norm

|

|

.

|

|

2

||.|{|_2}

induced by Tr, then

ρ
\rho

is closely related to a

∗
*

-homomorphism. In particular, if

ρ
\rho

is unital and approximately inner, then

ρ
\rho

is a

∗
*

-homomorphism of

M
M

into itself.
Decomposition of graded modules
Authors: Webb Cary
Published: June 2010

In this paper, the primary objective is to obtain decomposition theorems for graded modules over the polynomial ring

k
[
x
]

k[x]

, where

k
k

denotes a field. There is some overlap with recent work of Höppner and Lenzing. The results obtained include identification of the free, projective, and injective modules. It is proved that a module that is either reduced and locally finite or bounded below is a direct sum of cyclic submodules. Pure submodules are direct summands if they are bounded below. In such case, the pure submodule is itself a direct sum of cyclic submodules. It is also noted that Cohen and Gluck’s Stacked Bases Theorem remains true if the modules are graded.
The lower central series in some groups with the subnormal join property
Authors: Smith Howard
Published: June 2010

The following question is considered: What groups

G
G

are such that, given any subnormal subgroups

H
H

and

K
K

of

G
G

, with join

J
J

, and given any positive integers

a
a

and

b
b

, there exists an integer

c
c

such that

γ
c

(
J
)

{\gamma _c}(J)

is contained in

γ
a

(
H
)

γ
b

(
K
)

{\gamma _a}(H){\gamma _b}(K)

? It is shown that many, but not all, groups known to have the "subnormal join property" satisfy this further condition.
A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian
Authors: Roberts Paul C.
Published: June 2010

Let

R
R

be the polynomial ring

k
[
X
,
Y
,
Z
]

k[X,Y,Z]

localized at the maximal ideal

M
=
(
X
,
Y
,
Z
)

M = (X,Y,Z)

. We construct a prime ideal

P
P

in

R
R

which is equal to the ideal of

m
m

generic lines through the origin modulo

M
m

{M^m}

, and we show that, for suitable choice of

m
m

, the symbolic blow-up of such an ideal

P
P

is not Noetherian.
Consecutive primitive roots in a finite field. II
Authors: Cohen Stephen D.
Published: June 2010

The proof of the theorem that every finite field of order

q
(
>
3
)

q( > 3)

such that

q
≢
7
(

mod

12
)

q\not \equiv 7(\mod 12)

contains a pair of consecutive primitive roots is completed by consideration of the case in which

q
≡
1
(

mod

60
)

q \equiv 1(\mod 60)

.
A footnote to the multiplicative basis theorem
Authors: Gustafson William
Published: June 2010

We characterize those perfect fields

k
k

such that for each integer

n
⩾
1

n \geqslant 1

, but there are but finitely many isomorphism types of

k
k

-algebras of dimension

n
n

that are of finite representation type. Some remarks on the imperfect case are also presented.
Exponential sums of Lerch’s zeta functions
Authors: Wang Kai
Published: June 2010

For

x
x

not an an integer and

Re
⁡
(
s
)
>
0

\operatorname {Re} (s) > 0

, let
\[

F
(
x
,
s
)
=

∑

k
=
1

∞

e

2
π
i
k
x

k
s

F(x,s) = \sum \limits _{k = 1}^\infty {\frac {{{e^{2\pi ikx}}}}{{{k^s}}}}

\]
be the Lerch’s zeta function. In this note, we will show that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript gamma equals 1 Overscript m minus 1 Endscripts e Superscript minus 2 pi i gamma alpha slash m Baseline upper F left-parenthesis StartFraction gamma Over m EndFraction comma 1 minus n right-parenthesis equals StartFraction 1 Over n EndFraction left-parenthesis upper B Subscript n Baseline minus m Superscript n Baseline upper B Subscript n Baseline left-parenthesis StartFraction alpha Over m EndFraction minus left-bracket StartFraction alpha Over m EndFraction right-bracket right-parenthesis right-parenthesis">

∑

γ
=
1

m
−
1

e

−
2
π
i
γ
α

/

m

F

(

γ
m

,
1
−
n

)

=

1
n

(

B
n

−

m
n

B
n

(

α
m

−

[

α
m

]

)

)

\sum \limits _{\gamma = 1}^{m - 1} {{e^{ - 2\pi i\gamma \alpha /m}}} F\left ( {\frac {\gamma }{m},1 - n} \right ) = \frac {1}{n}\left ( {{B_n} - {m^n}{B_n}\left ( {\frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ]} \right )} \right )

\]
where

α
\alpha

is an integer and

α
≢
0
(

mod

m
)

\alpha \not \equiv 0(\mod m)

and

n
⩾
1

n \geqslant 1

. For

n
=
1

n = 1

, this formula is equivalent to the classical Eisenstein formula
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction alpha Over m EndFraction minus left-bracket StartFraction alpha Over m EndFraction right-bracket minus one half equals minus StartFraction 1 Over 2 m EndFraction sigma-summation Underscript gamma equals 1 Overscript m minus 1 Endscripts sine StartFraction 2 pi gamma alpha Over m EndFraction cotangent StartFraction pi gamma Over n EndFraction period">

α
m

−

[

α
m

]

−

1
2

=
−

1

2
m

∑

γ
=
1

m
−
1

sin
⁡

2
π
γ
α

m

cot
⁡

π
γ

n

.

\frac {\alpha }{m} - \left [ {\frac {\alpha }{m}} \right ] - \frac {1}{2} = - \frac {1}{{2m}}\sum \limits _{\gamma = 1}^{m - 1} {\sin \frac {{2\pi \gamma \alpha }}{m}} \cot \frac {{\pi \gamma }}{n}.

\]
Rotation invariant ideals in subalgebras of 𝐿^{∞}
Authors: Gorkin Pamela
Published: June 2010

In this paper, it is shown that the only (nontrivial) finitely generated rotation invariant ideals in

H
∞

{H^\infty }

are

z
n

H
∞

{z^n}{H^\infty }

for some positive integer

n
n

. Using results about function algebras, it is shown that other rotation invariant ideals exist. Rotation invariant ideals of other subalgebras of

L
∞

{L^\infty }

are also studied.
A 𝑞-polynomial identity
Authors: Yang Kung-Wei
Published: June 2010

We show that the

q
q

-polynomial coefficients

[

r

n
⋅
m

]

[_{\;r}^{n \cdot m}]

(see the definition preceding Theorem 1), which are the generating functions of the number of inversions between multisets, satisfy the beautiful identity
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript i equals 1 Overscript n Endscripts left-parenthesis sigma-summation Underscript j equals 0 Overscript m Endscripts q Superscript j m left-parenthesis i minus 1 right-parenthesis plus left-parenthesis Super Subscript 2 Superscript Super Superscript j Superscript right-parenthesis Baseline t Superscript j Baseline right-parenthesis equals sigma-summation Underscript r equals 0 Overscript n m Endscripts left-bracket Subscript r Baseline Superscript n dot m Baseline right-bracket q Superscript left-parenthesis Super Subscript 2 Superscript Super Superscript r Superscript right-parenthesis Baseline t Superscript r Baseline period">

∏

i
=
1

n

(

∑

j
=
0

m

q

j
m
(
i
−
1
)
+

(
2
j

)

t
j

)

=

∑

r
=
0

n
m

[

r

n
⋅
m

]

q

(
2
r

)

t
r

.

\prod \limits _{i = 1}^n {\left ( {\sum \limits _{j = 0}^m {{q^{jm(i - 1) + (_2^j)}}{t^j}} } \right ) = \sum \limits _{r = 0}^{nm} {[_{\;r}^{n \cdot m}} ]{q^{(_2^r)}}{t^r}} .

\]
Some sharp weak-type inequalities for holomorphic functions on the unit ball of 𝐶ⁿ
Authors: Tomaszewski Bogusław
Published: June 2010

Let

B
n

=
{
z
∈

C

n

:

|

z

|

>
1
}

{B^n} = \{ z \in {{\mathbf {C}}^n}:|z| > 1\}

,

S
n

=
∂

B
n

{S^n} = \partial {B^n}

and let

σ
n

{\sigma _n}

be the Haar measure on

S
n

{S^n}

. Then for all

f
∈

H
p

(
1
⩽
p
>
∞
)

f \in {H^p}(1 \leqslant p > \infty )

such that

Im
⁡
(
f
(
0
)
)
=
0

\operatorname {Im} (f(0)) = 0

and

t
>
0

t > 0

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript n Baseline left-parenthesis StartSet z element-of upper S Superscript n Baseline colon StartAbsoluteValue f left-parenthesis z right-parenthesis EndAbsoluteValue greater-than-or-slanted-equals t EndSet right-parenthesis less-than-or-slanted-equals upper C Subscript p Baseline dot StartFraction StartAbsoluteValue EndAbsoluteValue upper R e f StartAbsoluteValue EndAbsoluteValue Subscript p Superscript p Baseline Over t Superscript p Baseline EndFraction">

σ
n

(
{
z
∈

S
n

:

|

f
(
z
)

|

⩾
t
}
)
⩽

C
p

⋅

|

|

Re

f

|

|

p
p

t
p

{\sigma _n}(\{ z \in {S^n}:|f(z)| \geqslant t\} ) \leqslant {C_p} \cdot \frac {{||\operatorname {Re} \,f||_p^p}} {{{t^p}}}

\]
for some constant

C
p

{C_p}

depending only on

p
p

. The best constant

C
p

{C_p}

is found for

1
⩽
p
⩽
2

1 \leqslant p \leqslant 2

.
Some elements in the stable homotopy of spheres
Authors: Lin Wên Hsiung
Published: June 2010

An infinite family in

2

π
∗
s

{}_2\pi _*^s

at Adams filtration 4 is constructed.
Continuity of the inverse
Authors: Pfister Helmut
Published: June 2010

We present a simple device for proving the continuity of the inverse in a group with a locally Čech-complete topology which makes the multiplication continuous; our proof even works in case the topology is regular and locally strongly countably complete.
Bounds on the maximum number of vectors with given scalar products
Authors: Deza M. , Frankl P.
Published: June 2010

Suppose

M
M

,

L
L

are sets of real numbers,

V
=
{

υ
1

,
…
,

υ
m

}

V = \{ {\upsilon _1}, \ldots ,{\upsilon _m}\}

is a collection of vectors in

R
n

{R^n}

, having

k
k

nonzero coordinates all from

M
M

and satisfying

(

υ
i

,

υ
j

)
∈
L

({\upsilon _i},{\upsilon _j}) \in L

for

i
≠
j

i \ne j

. Theorem 1.1 establishes a polynomial upper bound for

|
V
|

\left | V \right |

, generalizing previous results for subsets of a set and

(
0
,
±
1
)

(0, \pm 1)

-vectors. Theorem 1.4 asserts that if

|
L
|

=
s

\left | L \right | = s

then

|
V
|

⩽

(

n
+
s

s

)

\left | V \right | \leqslant \left ( {\begin {array}{*{20}{c}} {n + s} \\ s \\ \end {array} } \right )

. For

M
=
{
−
1
,
1
}

M = \{ - 1,1\}

,

L
=
[
−
(
k
−
1
)
,
k
−
1
]

L = [ - (k - 1),k - 1]

, Theorem 1.5 gives

|
V
|

⩽

2

k
−
1

(

n

k
−
1

)

/

k

\left | V \right | \leqslant {2^{k - 1}}\left ( {\begin {array}{*{20}{c}} n \\ {k - 1} \\ \end {array} } \right )/k

, where equality holds if and only if

V
V

is a "signed"

(
n
,
k
,
k
−
1
)

(n,k,k - 1)

Steiner-system.
Extreme points in convex sets of symmetric matrices
Authors: Ycart Bernard
Published: June 2010

This paper deals with the following problem: What are the extreme points of a convex set

K
K

of

n
×
n

n \times n

matrices, which is the intersection of the set

S
n

{S_n}

of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices

H
?

H?

? In the case where the facial structure of

H
H

is known, we expose a general method to determine the extreme points of

K
K

(Theorem 1). Then, we apply this method to the set of correlation matrices, characterizing its extreme points in Theorem 2, which is our main theorem. A corollary describes thoroughly the extreme points of rank 2.
Affine endomorphisms with a dense orbit
Authors: Kasuga Tatsuro
Published: June 2010

For a continuous endomorphism

f
f

on a locally compact group

X
X

and

a
∈
X

a \in X

, we define an affine endomorphism

f
a

:
X
→
X

{f_a}:X \to X

. We prove that if

f
a

{f_a}

is not one to one and if

(

X
,

f
a

)

\left ( {X,{f_a}} \right )

has a dense orbit then

X
X

is compact.
A note on module homotopy and chain homotopy
Authors: Hardie Keith , Hilton Peter
Published: June 2010

The Eckmann-Hilton (injective) homotopy category of modules over a ring

Λ
\Lambda

is equivalent to a certain full subcategory of the cochain homotopy category of cochain complexes of modules over

Λ
\Lambda

.
Uniform distribution of second-order linear recurring sequences
Authors: Turnwald G.
Published: June 2010

A complete classification is obtained for all second-order linear recurring sequences uniformly distributed modulo an ideal of a Dedekind domain.
On a conjecture of Kátai concerning weakly composite numbers
Authors: Galambos Janos
Published: June 2010

A number is called weakly composite if the sum of the reciprocals of its prime divisors is bounded by two. In this note it is proved that, forn⩾n0n \geqslant {n_0}, there is a weakly composite number betweennnandn+log⁡log⁡log⁡nn + \log \log \log n.
Dirichlet integral and star-function inequalities
Authors: Quine J. R.
Published: June 2010

Let

λ
(
z
)

\lambda (z)

be a smooth function in an annulus,

λ
~

(
z
)

\tilde \lambda (z)

its symmetric rearrangement and

λ
∗

(
z
)

{\lambda ^ * }(z)

its star-function. A formula is proved relating

Δ

λ
∗

,
Δ
λ

\Delta {\lambda ^ * },\Delta \lambda

, and the Dirichlet integrands of

λ
\lambda

and

λ
~

\tilde \lambda

. The formula shows the relationship between Dirichlet integral inequalities and the subharmonicity of

λ
∗

{\lambda ^ * }

for subharmonic

λ
\lambda

, and gives an explicit formula for

Δ

λ
∗

\Delta {\lambda ^ * }

.
Metrizability of general ANR
Authors: Tsuda Kōichi
Published: June 2010

We show that every nonmetrizable

ANR(

P

)

{\text {ANR(}}\mathcal {P}{\text {)}}

contains a copy of a Tychonoff cube of uncountable weight. Hence, every finite dimensional

ANR(

P

)

{\text {ANR(}}\mathcal {P}{\text {)}}

is metrizable, and every

ANR(

P

)

{\text {ANR(}}\mathcal {P}{\text {)}}

, each point of which is a

G
δ

{G_\delta }

-set, is metrizable, where

P

\mathcal {P}

denotes the class of all paracompact

p
p

-spaces.
Remarks on Pettis integrability
Authors: Huff R.
Published: June 2010

Characterizations of Pettis integrability, including the Geitz-Talagrand core theorem, are derived in an easy way.
An open book decomposition for 𝑅𝑃²×𝑆¹
Authors: Klassen Eric
Published: June 2010

In this paper an open book decomposition for

R

P
2

×

S
1

R{P^2} \times {S^1}

is exhibited.
Coincidence theorem and saddle point theorem
Authors: Komiya H.
Published: June 2010

We discuss Browder’s coincidence theorem and derive a saddle point theorem from it.
Definable principal congruences and 𝑅-stable identities
Authors: Simons G. E.
Published: June 2010

We show that an algebra over an infinite field generates a variety with definable principal congruences if and only if it is commutative. A similar result is proved for polynomial rings. The main tool used is the notion from the theory of PI-rings of an

R
R

-stable identity.
Selfadjoint nonoscillatory second order linear 𝐵*-algebra differential equations
Authors: Lopez Manuel
Published: June 2010

The main result in this paper states that the second order linear

B
∗

{B^*}

-algebra differential equation

(
p
(
t
)

y
′

)
+
q
(
t
)
y
=
0

(p(t)y’) + q(t)y = 0

, where

p
(
t
)

p(t)

is positive and

q
(
t
)

q(t)

is Hermitian for each

t
t

, is nonoscillatory on

[

t
0

,
∞
)

[{t_0},\infty )

if the scalar equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-vertical-bar p Superscript negative 1 Baseline left-parenthesis t right-parenthesis double-vertical-bar Superscript negative 1 Baseline upper W prime right-parenthesis prime plus double-vertical-bar q left-parenthesis t right-parenthesis double-vertical-bar upper W equals 0">

(

‖

p

−
1

(
t
)

‖

−
1

W
′

)
′

+

‖

q
(
t
)

‖

W
=
0

({\left \| {{p^{ - 1}}(t)} \right \|^{ - 1}}W’)’ + \left \| {q(t)} \right \|W = 0

is nonoscillatory on

[

t
0

,
∞
)

[{t_0},\infty )

. Consequently, every criterion on nonoscillation in the scalar case automatically produces another one in the

B
∗

{B^*}

-algebra case.
An extremal problem for polynomials with a prescribed zero
Authors: Milovanović Igor Ž.
Published: June 2010

In this paper a new elementary proof for solving one extremal problem of real polynomials with a given real zero is given.
The normal extensions of subgroup topologies
Authors: Clark Bradd , Schneider Victor
Published: June 2010

Let

H
H

be a topological group contained in a group

G
G

. A topology which makes

G
G

a topological group inducing the given topology on

H
H

is called an extending topology. The set of all extending topologies forms a complete semilattice in the lattice of group topologies on

G
G

. The structure of this semilattice is studied by considering normal subgroups which intersect

H
H

in the identity.
Twisted sums of Banach and nuclear spaces
Authors: Domański Paweł
Published: June 2010

A twisted sum of (topological vector) spaces

Y
Y

and

Z
Z

is a space

X
X

with a subspace

Y
1

{Y_1}

isomorphic to

Y
Y

for which

X

/

Y
1

X/{Y_1}

is isomorphic to

Z
Z

. It splits if

Y
1

{Y_1}

is complemented. It is proved that every twisted sum of a Banach space

Y
Y

and a nuclear space

Z
Z

splits. Köthe sequence spaces

Z
Z

for which this holds are characterized. Every locally convex twisted sum of a nuclear Fréchet space

Y
Y

and a Banach space

Z
Z

splits too. If

Z
Z

is superreflexive, then the local convexity assumption on the twisted sum may be omitted. Other results of this kind on Köthe sequence spaces are obtained.
Long periodic orbits of the triangle map
Authors: Scarowsky Manny , Boyarsky Abraham
Published: June 2010

Let

τ
:
[
0
,
1
]
→
[
0
,
1
]

\tau :[0,1] \to [0,1]

be defined by

τ
(
x
)
=
2
x

\tau (x) = 2x

on

[
0
,
1

/

2
]

[0,1/2]

and

τ
(
x
)
=
2
(
1
−
x
)

\tau (x) = 2(1 - x)

on

[
1

/

2
,
1
]

[1/2,1]

. We consider

τ
\tau

restricted to the domain
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript upper N Baseline equals StartSet 2 a slash p Superscript upper N Baseline comma upper N greater-than-or-slanted-equals 1 comma 0 less-than-or-slanted-equals 2 a less-than-or-slanted-equals p Superscript upper N Baseline comma left-parenthesis a comma p right-parenthesis equals 1 EndSet">

D
N

=
{
2
a

/

p
N

,
N
⩾
1
,
0
⩽
2
a
⩽

p
N

,
(
a
,
p
)
=
1
}

{D_N} = \{ 2a/{p^N},N \geqslant 1,0 \leqslant 2a \leqslant {p^N},(a,p) = 1\}

where

p
p

is any odd prime. Let

k
⩾
1

k \geqslant 1

be the minimum integer such that

p
N

|

2
k

±
1

{p^N}|{2^k} \pm 1

. Then there are

(
(

p

−
1
)
⋅

p

N
−
1

)

/

2
k

(({\text {p}} - 1) \cdot {p^{N - 1}})/2k

periodic orbits of

τ

|

D
N

\tau {|_{{D_N}}}

, having equal length, and there are

k
k

points in each orbit. Furthermore, the proportion of points in any of these periodic orbits which lie in an interval

(
c
,
d
)

(c,d)

approaches

d
−
c

d - c

as

p

N
−
1

→
∞

{p^{N - 1}} \to \infty

. An application to the irreducibility of certain nonnegative matrices is given.
A Wiener inversion-type theorem
Authors: Holub James R.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W left-parenthesis upper D right-parenthesis equals StartSet f left-parenthesis z right-parenthesis equals normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline a Subscript n Baseline z Superscript n Baseline StartAbsoluteValue EndAbsoluteValue StartAbsoluteValue f EndAbsoluteValue vertical-bar Subscript 1 Baseline equals normal upper Sigma Subscript n equals 0 Superscript normal infinity Baseline StartAbsoluteValue a Subscript n Baseline EndAbsoluteValue greater-than plus normal infinity EndSet">

W
(
D
)
=
{
f
(
z
)
=

Σ

n
=
0

∞

a
n

z
n

|

|

|

f

|

|

1

=

Σ

n
=
0

∞

|

a
n

|

>
+
∞
}

W(D) = \{ f(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}|\;||f|{|_1} = \Sigma _{n = 0}^\infty |{a_n}| > + \infty \}

,

f
(
z
)

f(z)

a function in

W
(
D
)

W(D)

for which

f
(
0
)
=
1

f(0) = 1

, and

M
f

{M_f}

the operator of multiplication by

f
(
z
)

f(z)

on

W
(
D
)

W(D)

. It is shown that if

k
k

and

m
m

are integers for which

0
≤
m
≤
k
−
1

0 \leq m \leq k - 1

and

X
k
m

X_k^m

is the closed subspace of

W
(
D
)

W(D)

spanned by

{

z

n
k
+
i

|

n
=
0
,
1
,
…
;
i
=
0
,
1
,
…
,
m
}

\{ {z^{nk + i}}|n = 0,1, \ldots ;i = 0,1, \ldots ,m\}

, then

M
f

{M_f}

is bounded below on

X
k
m

⇔
f
(
z
)

X_k^m \Leftrightarrow f(z)

does not have

k
−
m

k - m

distinct zeros in any set of the form

{

w
i

z
0

|

0
≤
i
≤
k
−
1
;

|

z
0

|

=
1
}

\{ {w^i}{z_0}|0 \leq i \leq k - 1;|{z_0}| = 1\}

, where

w
w

is a primitive

k
k

th root of unity.
The group generated by unipotent operators
Authors: Fong C. K. , Sourour A. R.
Published: June 2010

The group generated by unipotent

n
×
n

n \times n

complex matrices is

S

L

n

(
C
)

{\text {S}}{{\text {L}}_n}(C)

, and every member of the latter is a product of three unipotent matrices. The group generated by unipotent operators on Hilbert space

H

\mathcal {H}

is

GL(

H

)

{\text {GL(}}\mathcal {H}{\text {)}}

, and every invertible operator is a product of six unipotent operators of order 2.
On the heat kernel comparison theorems for minimal submanifolds
Authors: Markvorsen Steen
Published: June 2010

In [3], Cheng, Li and Yau proved comparison theorems (upper bounds) for the heat kernels on minimal submanifolds of space forms. In the present note we show that these comparison theorems together with a series of corollaries remain true for minimal submanifolds in ambient spaces with just an upper bound on the sectional curvature.
Decompositions of spaces determined by compact subsets
Authors: Tanaka Yoshio
Published: June 2010

Let

X
X

be a

k
′

k’

-space, and let

F

\mathcal {F}

be a closed cover of (locally) compact subsets of

X
X

. Then

X
X

is decomposed into a closed discrete subset and a locally compact subset if

X
X

is dominated by

F

\mathcal {F}

, or

X
X

has the weak topology with respect to a point-countable cover

F

\mathcal {F}

. Here, a cover of a space is point-countable if every point is in at most countably many elements of the cover.
A counterexample to a result concerning controlled approximation
Authors: Jia Rong Qing
Published: June 2010

A result of Strang and Fix states that if the order of controlled approximation from a collection of locally supported elements is

k
k

, then there is a linear combination

Ω
\Omega

of those elements and their translates such that any polynomial of degree less than

k
k

can be reproduced by

Ω
\Omega

and its translates. This paper gives a counterexample to their result.
Series that converge on sets of null density
Authors: Estrada R. , Kanwal R. P.
Published: June 2010

It is shown that a series of positive terms that converges on all sets of null density should be convergent. Using this result we construct examples of complete topological vector spaces that are proper subspaces of a Banach space, but whose dual spaces coincide with the dual space of the Banach space.
Selection theorems and invariance of Borel pointclasses
Authors: Bhattacharya Sreela , Srivastava S. M.
Published: June 2010

We generalize some known selection theorems and give simple proofs of results on the invariance of Borel pointclasses obtained by Saint-Raymond, Jayne and Rogers, and Kunen and Miller.
A best constant and the Gaussian curvature
Authors: Hong Chong Wei
Published: June 2010

For axisymmetric

f
∈

C
∞

(

S
2

)

f \in {C^\infty }({S^2})

we find conditions to make

f
f

the scalar curvature of a metric pointwise conformal to the standard metric of

S
2

{S^2}

. Closely related to these results, we prove that in the inequality (Moser [8])
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper S squared Endscripts e Superscript u Baseline less-than-or-equal-to upper C e Superscript double-vertical-bar nabla u double-vertical-bar Super Subscript 2 Super Superscript 2 Superscript slash 16 pi Baseline for-all u element-of upper H 1 squared left-parenthesis upper S squared right-parenthesis with integral Underscript upper S squared Endscripts u equals 0 comma">

∫

S
2

e
u

≤
C

e

‖

∇
u

‖

2
2

/

16
π

∀
u
∈

H
1
2

(

S
2

)

 with 

∫

S
2

u
=
0

,

\int _{{S^2}} {{e^u} \leq C{e^{\left \| {\nabla u} \right \|_2^2/16\pi \quad }}\forall u \in H_1^2({S^2})} {\text { with }}\int _{{S^2}} {u = 0} ,

\]
, the best constant

C
=

Vol(

S

2

)

C = {\text {Vol(}}{{\text {S}}^2}{\text {)}}

.
Maps in 𝑅ⁿ with finite-to-one extensions
Authors: Starbird Michael
Published: June 2010

Suppose

f
:
X
→

R

n

f:X \to {{\mathbf {R}}^n}

is a continuous function from a closed subset

X
X

of

R

n

{{\mathbf {R}}^n}

into

R

n

{{\mathbf {R}}^n}

. The Tietze Extension Theorem states that there is a continuous function

F
:

R

n

→

R

n

F:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n}

that extends

f
f

. Here we consider the question of when the extension

F
F

can be chosen with

F

|

R

n

−
X

F|{{\mathbf {R}}^n} - X

being finite-to-one. Not every map

f
f

has such an extension. If

f
(
X
)

f(X)

is sufficiently nice, then there is such a finite-to-one extension. For example, it is shown that if

f
:
X
→

R

n

f:X \to {{\mathbf {R}}^n}

is a map and

f
(
X
)
⊂

R

n
−
1

×
{
0
}

f(X) \subset {{\mathbf {R}}^{n - 1}} \times \{ 0\}

then there is a continuous extension

F
:

R

n

→

R

n

F:{{\mathbf {R}}^n} \to {{\mathbf {R}}^n}

such that

F

|

R

n

−
X

F|{{\mathbf {R}}^n} - X

is finite-to-one. On the other hand, if

X
X

is nowhere dense and

f
(
X
)

f(X)

contains an open set, then there definitely is not such a finite-to-one extension. Other examples and theorems show that the finite-to-one extendability of a map

f
:
X
→

R

n

f:X \to {{\mathbf {R}}^n}

is not necessarily a function of the topology of

f
(
X
)

f(X)

, but may depend on its embedding or on the map

f
f

.
Some dimension 3 cases of the canonical element conjecture
Authors: Huneke Craig , Koh Jee
Published: June 2010

The Canonical Element Conjecture holds for those

3
3

-dimensional local rings of depth 2 whose second local cohomology module is a finite direct sum of cyclic modules, and for those

3
3

-dimensional local rings whose second local cohomology module is cyclic.
A sofic system with infinitely many minimal covers
Authors: Williams Susan
Published: June 2010

We give an example of a sofic system

S
S

together with an infinite collection of nonconjugate subshift of finite type covers of

S
S

such that none of these covers factors through another subshift-of-finite-type cover to which it is not topologically conjugate.
Jacobson radical of filtered algebras
Authors: Behr Erazm J.
Published: June 2010

Using elementary graded ring theory methods we show that the Jacobson radical of certain filtered algebras is zero. We then use this to propose a simpler and more general proof of one of the main results contained in [1].
Approximation by rational functions
Authors: DeVore Ronald A.
Published: June 2010

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If

f
′

f’

is in

L
log
⁡
L

L\log L

on a finite interval, then

f
f

can be approximated in the uniform norm by rational functions of degree

n
n

to an error

O
(
1

/

n
)

O(1/n)

on that interval.
The norm exponent in Galois extensions of number fields
Authors: Opolka Hans
Published: June 2010

The purpose of this note is to provide an upper bound for the exponent of the norm residue group

k
∗

/

Norm

K

/

k

⁡
(

K
∗

)

{k^ * }/\operatorname {Norm}_{K/k}({K^ * })

of a finite Galois extension

K

/

k

K/k

of number fields in terms of a certain central extension of

K

/

k

K/k

.
A unified approach to uniqueness of Walsh series and Haar series
Authors: Wade William R.
Published: June 2010

We obtain a uniqueness theorem for Walsh series valid for subsequences of

2
n

{2^n}

th partial sums which satisfy a pointwise growth condition.
On unilateral and bilateral 𝑛th Peano derivatives
Authors: Laczkovich M. , Preiss D. , Weil C.
Published: June 2010

A valid proof is given of the assertion that an

n
n

th Peano derivative that is allowed to attain infinite values is still a function of Baire class one. Also, it is shown that a finite, unilateral

n
n

th Peano derivative is a function of Baire class one. Finally, an example is given that if infinite values are allowed (actually just

+
∞

+ \infty

) a unilateral

n
n

th Peano derivative need not be of Baire class one.
Finite generation of certain subrings
Authors: Fogarty John
Published: June 2010

A more geometric approach can be used to prove finite generation of certain subrings, notably invariants under reductive group actions.
On the cohomology of associative algebras and Lie algebras
Authors: Farnsteiner Rolf
Published: June 2010

This paper presents a sufficient condition for the vanishing of the cohomology groups of an associative algebra.
On the ergodic Hilbert transform for Lamperti operators
Authors: Sato Ryotaro
Published: June 2010

This paper is devoted to the proof of almost everywhere existence of the ergodic Hilbert transform for a class of Lamperti operators.
Riesz theory without axiom of choice
Authors: Martensen Erich
Published: June 2010

In this paper the Riesz theory for compact linear operators in a normed vector space is considered from the point of view of how far the axiom of choice is involved. Special attention is drawn to the theorem, by which for the operator

I
−
A
,
A

I - A,A

being compact, the index vanishes and the nullspace has a closed algebraic complement. It is shown that this can be proved without making use of the axiom of choice.
A note on the 𝑏𝑃-component of (4𝑛-1)-dimensional homotopy spheres
Authors: Stolz Stephan
Published: June 2010

The

b
P

bP

-component of a

(

4
n
−
1

)

\left ( {4n - 1} \right )

-dimensional homotopy sphere

Σ
∈

θ

4
n
−
1

≅
b

P

4
n

⊕

(

Co

ker
⁡
J

)

4
n
−
1

\Sigma \in {\theta _{4n - 1}} \cong b{P_{4n}} \oplus {\left ( {{\text {Co}}\ker J} \right )_{4n - 1}}

bounding a spin manifold

M
M

is shown to be computable in terms of the signature and the decomposable Pontrjagin numbers of

M
M

.
Entropies of automorphisms of a topological Markov shift
Authors: Lind D. A.
Published: June 2010

Let

σ
\sigma

be a mixing topological Markov shift,

λ
\lambda

a weak Perron number,

q

(
t
)

q\left ( t \right )

a polynomial with nonnegative integer coefficients, and

r
r

a non-negative rational. We construct a homeomorphism commuting with

σ
\sigma

whose topological entropy is

log
⁡

[

q

(
λ
)

q

(

1

/

λ

)

]

r

\log {\left [ {q\left ( \lambda \right )q\left ( {1/\lambda } \right )} \right ]^r}

. These values are shown to include the logarithms of all weak Perron numbers, and are dense in the nonnegative reals.
Ideals containing monics
Authors: Nashier Budh , Nichols Warren
Published: June 2010

If an ideal

I
I

of

R
[
X
]

R[X]

contains a monic, then every monic in

I
I

modulo

J
R
[
X
]

JR[X]

(

J
J

being an ideal contained in the Jacobson radical of

R
R

) can be lifted to a monic in

I
I

. This result is used to give an elementary proof of Horrocks’ Theorem.
The determination of a parabolic equation from initial and final data
Authors: Rundell William
Published: June 2010

It is shown that an unknown, spatially-dependent coefficient

a
(
x
)

a(x)

in the parabolic equation

u
t
−
Δ
u
+
a
(
x
)
u
=
0

ut - \Delta u + a(x)u = 0

can be determined from a knowledge of both initial and final data. An existence and uniqueness theorem is given and the continuous dependence of the function

a
(
x
)

a(x)

on the data is examined.
Asymptotically periodic solutions of a class of second order nonlinear differential equations
Authors: Liang Zhong Chao
Published: June 2010

In this paper we give necessary and sufficient conditions for all solutions of the system
\[

(

S

)

x
′

=
y
,

y
′

=
−
a
(
t
)
f
(
x
)
g
(
y
)

({\text {S}})\quad x’ = y,\quad y’ = - a(t)f(x)g(y)

\]
to be oscillatory or bounded, for all orbits of the system
\[

(

S

1

)

x
′

=
y
,

y
′

=
−
α
f
(
x
)
g
(
y
)

({{\text {S}}_1})\quad x’ = y,\quad y’ = - \alpha f(x)g(y)

\]
to be periodic, where

a
(
t
)
→
α
>
0

a(t) \to \alpha > 0

as

t
→
∞

t \to \infty

, and for every orbit of (S) to approach a periodic orbit of (S

1

_{1}

). The conditions assuring that every solution of (S) is asymptotically periodic are also established.
Complex interpolation and convexity
Authors: Vignati Marco
Published: June 2010

The relations between the complex theory of interpolation for families of Banach spaces and the notion of uniform convexity are studied. It is proven that the moduli of uniform convexity vary smoothly with the interpolation spaces. A new notion of "distance" between Banach spaces is introduced.
Linear sums of certain analytic functions
Authors: Singh Ram , Paul Surinder
Published: June 2010

Let

f
f

belong to a certain subclass of the class of functions which are regular in the unit disc

E
=
{
z
:

|

z

|

>
1
}

E = \{ z:|z| > 1\}

. Suppose that

ϕ
=
ϕ
(
f
,

f
′

,
f
)

\phi = \phi (f,f’,f)

and

ψ
=
ψ
(
f
,

f
′

,
f
)

\psi = \psi (f,f’,f)

are regular in

E
E

with

Re
⁡
ϕ
>
0

\operatorname {Re} \phi > 0

in

E
E

and

Re
⁡
ψ
≯
0

\operatorname {Re} \psi \ngtr 0

in the whole of

E
E

. In this paper we consider the following two new types of problems: (i) To find the ranges of the real numbers

λ
\lambda

and

μ
\mu

such that

Re
⁡
(
λ
ϕ
+
μ
ψ
)
>
0

\operatorname {Re} (\lambda \phi + \mu \psi ) > 0

in

E
E

. (ii) To determine the largest number

ρ
,
0
>
ρ
>
1

\rho ,0 > \rho > 1

, such that

Re
⁡
(
ϕ
+
ψ
)
>
0

\operatorname {Re} (\phi + \psi ) > 0

in

|

z

|

>
ρ

|z| > \rho

.
A companion to the Oseledec multiplicative ergodic theorem
Authors: Watkins Joseph C.
Published: June 2010

Let

F
1

,

F
2

,
…

{F_1},{F_2}, \ldots

be a stationary sequence of continuously differentiable mappings from

[
0
,
1
]

[0,1]

into the set of

d
×
d

d \times d

matrices. Assume

F
k

(
0
)
=
I

{F_k}(0) = I

for each

k
k

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-bracket sup Underscript 0 less-than-or-equal-to p less-than-or-equal-to 1 Endscripts StartAbsoluteValue EndAbsoluteValue upper F prime Subscript k Baseline left-parenthesis p right-parenthesis StartAbsoluteValue EndAbsoluteValue right-bracket greater-than normal infinity">

E
[

sup

0
≤
p
≤
1

|

|

F
k
′

(
p
)

|

|

]
>
∞

E[{\sup _{0 \leq p \leq 1}}||{F’_k}(p)||] > \infty

. Let

I

\mathcal {I}

denote the invariant sigma field for the sequence. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts upper F Subscript n Baseline left-parenthesis StartFraction 1 Over n EndFraction right-parenthesis midline-horizontal-ellipsis upper F 2 left-parenthesis StartFraction 1 Over n EndFraction right-parenthesis upper F 1 left-parenthesis StartFraction 1 Over n EndFraction right-parenthesis equals exp upper E left-bracket upper F prime 1 left-parenthesis 0 right-parenthesis vertical-bar script upper I right-bracket">

lim

n
→
∞

F
n

(

1
n

)

⋯

F
2

(

1
n

)

F
1

(

1
n

)

=
exp
⁡
E
[

F
1
′

(
0
)

|

I

]

\lim \limits _{n \to \infty } {F_n}\left ( {\frac {1}{n}} \right ) \cdots {F_2}\left ( {\frac {1}{n}} \right ){F_1}\left ( {\frac {1}{n}} \right ) = \exp E[{F’_1}(0)|\mathcal {I}]

\]
with probability one.
A generalization of the flat cone condition for regularity of solutions of elliptic equations
Authors: Lieberman Gary M.
Published: June 2010

Barrier arguments are used to prove regularity of boundary points for a large class of uniformly elliptic operators when the domain satisfies a geometric condition. The condition is that the exterior of the domain contains a suitable lower dimensional set.
Daugavet’s equation and operators on 𝐿¹(𝜇)
Authors: Holub James R.
Published: June 2010

Generalizing a result of Babenko and Pichugov, it is shown that if

T
T

is a weakly compact operator on

L
1

(
μ
)

{L^1}(\mu )

, where

μ
\mu

is a

σ
\sigma

-finite nonatomic measure, then

‖

I
+
T

‖

=
1
+

‖
T
‖

\left \| {I + T} \right \| = 1 + \left \| T \right \|

. A characterization of all operators

T
T

on

L
1

(
μ
)

{L^1}(\mu )

having this property is also given.
Decidability of relation algebras with weakened associativity
Authors: Németi I.
Published: June 2010

Tarski showed that mathematics can be built up in the equational theory EqRA of relation algebras (RA’s), hence EqRA is undecidable. He raised the problem "how much associativity of relation composition is needed for this result." Maddux defined the classes

NA
⊃
WA
⊃
SA
⊃
RA

\operatorname {NA} \supset \operatorname {WA} \supset \operatorname {SA} \supset \operatorname {RA}

by gradually weakening the associativity of relation composition, and he proved that the equational theory of SA is still undecidable. We showed, elsewhere, that mathematics can be built up in SA, too. In the present paper we prove that the equational theories of WA and NA are already decidable. Hence mathematics cannot be built up in WA or NA. This solves a problem in the book by Tarski and Givant.
Polynomials of 2-cable-like links
Authors: Lickorish W. B. R. , Lipson A. S.
Published: June 2010

Morton and Short [MS] have established experimentally that two knots

K
1

{K_1}

and

K
2

{K_2}

may have the same

2
2

-variable polynomial

P
(
l
,
m
)

P(l,m)

(see [FYHLMO], [LM]) while

2
2

-cables on

K
1

{K_1}

and

K
2

{K_2}

can be distinguished by

P
P

. We prove here that if

K
1

{K_1}

and

K
2

{K_2}

are a mutant pair, then their

2
2

-cables and doubles (and other satellites which are

2
2

-stranded on the boundary of the mutating tangle) cannot be distinguished by

P
P

. Similar results are true for the unoriented knot polynomial

Q
Q

and its oriented two-variable counterpart

F
F

(see [BLM], [K]). The results are false if

K
1

,

K
2

{K_1},{K_2}

are links of more than one component.
Cubic character sums of cubic polynomials
Authors: Wright David J.
Published: June 2010

A complete evaluation is given of the sum over an arbitrary finite field of the values of a nontrivial cubic character of the field applied to an arbitrary polynomial of degree not greater than three in one variable defined over the field. Previous evaluations for the fields of prime order were given in theses of Friedman and Lagarias. The evaluation given below makes simple use of standard facts about equivalence of binary cubic forms. An interesting consequence of this evaluation is given connecting the values of these sums over the space of all polynomials of degree not greater than three over the finite field. The evaluation of these sums is of relevance to the theory of Shintani’s Dirichlet series associated to the space of binary cubic forms since they appear in the residues of the cubic twists of these series.
Minimality of geodesics in Grassmann manifolds
Authors: Porta Horacio , Recht Lázaro
Published: June 2010

In the Grassmann manifold of an arbitrary

C
∗

{C^ * }

-algebra, the geodesics of length less than

π
\pi

are curves of minimal length.
On the purely inseparable closure of rings
Authors: Sato Shizuka
Published: June 2010

Let

K
⫅
R

K \subseteqq R

be commutative rings with identity 1. Let

D
=
{

D
i

}

D = \{ {D_i}\}

be a higher derivation of

R
R

. We shall prove in this paper that if

K
K

is invariant with respect to

D
D

, the purely inseparable closure

K
R

¯

\overline {{K_R}}

of

K
K

in

R
R

is invariant with respect to

D
D

and the formal power series ring

K
R

¯

[
[
t
]
]

\overline {{K_R}} [[t]]

is purely inseparably closed in

R
[
[
t
]
]

R[[t]]

.
On the degree of the Brandt-Lickorish-Millett-Ho polynomial of a link
Authors: Kidwell Mark E.
Published: June 2010

Let

Q
L

{Q_L}

be the link polynomial defined by Brandt, Lickorish, Millett, and Ho. Let

deg
⁡

Q
L

\deg {Q_L}

be the maximum degree of a nonzero term. If

p
(
L
)

p(L)

is any regular link projection and

B
B

is any bridge (maximal connected component after undercrossing points are deleted), define the length of

B
B

as the number of crossings in which the overcrossing segment is a part of

B
B

. Theorem 1. Let

p
(
L
)

p(L)

be a connected, regular link projection with

N
N

crossing points. Let

K
K

be the maximal length of any bridge in

p
(
L
)

p(L)

. Then

deg
⁡

Q
L

≤
N
−
K

\deg {Q_L} \leq N - K

. Theorem 2. If

p
(
L
)

p(L)

is a prime, connected alternating projection with

N
>
0

N > 0

crossing points, then the coefficient of

x

N
−
1

{x^{N - 1}}

is a positive number.
On locally repeated values of certain arithmetic functions. III
Authors: Erdős Paul , Pomerance Carl , Sárközy András
Published: June 2010

Let

d
(
n
)

d(n)

denote the number of natural divisors of

n
n

. It is shown that the number of

n
≤
x

n \leq x

with

d
(
n
)
=
d
(
n
+
1
)

d(n) = d(n + 1)

is

O
(
x

/

log
⁡
log
⁡
x

)

O(x/\sqrt {\log \log x} )

. In addition, certain related problems and results are presented.
The generalized Burgers’ equation and the Navier-Stokes equation in 𝑅ⁿ with singular initial data
Authors: Avrin Joel D.
Published: June 2010

From an abstract theory of Weissler we construct a simple local existence theory for a generalization of Burgers’ equation and the Navier-Stokes equation in the Banach space

L
p

(

R

n

)

{L^p}({{\mathbf {R}}^n})

. Our conditions on

p
p

recover the conditions of Giga and Weissler in the latter case except for the borderline situation

p
=
n

p = n

. For the generalized Burgers’ equation our results are apparently new; moreover we show that these local solutions are in fact global solutions in this case. We also obtain results for the generalized Burgers’ equation with

R

n

{{\mathbf {R}}^n}

replaced by a bounded domain

Ω
\Omega

with smooth boundary. Using a somewhat more complex abstract theory of Weissler, we are able to improve on our results found in the case

Ω
=

R

n

\Omega = {{\mathbf {R}}^n}

, and also obtain global existence.
Chaos and higher order differences
Authors: Neuberger John W.
Published: June 2010

Asymptotic doubling of magnitude of higher order differences of a function is linked with abrupt character changes in the function.
Homotopy theories for diagrams of spaces
Authors: Dror Farjoun E.
Published: June 2010

We show that the category of diagrams of topological spaces (or simplicial sets) admits many interesting model category structures in the sense of Quillen [8]. The strongest one renders any diagram of simplicial complexes and simplicial maps between them both fibrant and cofibrant. Namely, homotopy invertible maps between such are the weak equivalences and they are detectable by the "spaces of fixed points." We use a generalization of the method for defining model category structure of simplicial category given in [5].
On 𝐿₁-contraction for systems of conservation laws
Authors: Patiño Jorge G. S.
Published: June 2010

We prove that for

2
×
2

2 \times 2

, strictly hyperbolic, genuinely nonlinear systems of conservation laws, there is no metric

D
D

such that
\[

∫

−
∞

∞

D
(
u
(
x
,
t
)
,
c
)
d
x

\int _{ - \infty }^\infty {D(u(x,t),c)dx}

\]
is a nonincreasing function of time for every weak solution

u
,

u
0

(
±
∞
)
=
c

u,{u_0}( \pm \infty ) = c

.
On the Orlicz-Pettis property in nonlocally convex 𝐹-spaces
Authors: Nawrocki M.
Published: June 2010

Recently, J. H. Shapiro showed that, contrary to the case of separable

F
F

-spaces with separating duals, the Orlicz-Pettis theorem fails for

h
p

,
0
>
p
>
1

{h_p},0 > p > 1

, and some other nonseparable

F
F

-spaces of harmonic functions. In this paper we give new, much simpler examples of

F
F

-spaces for which the Orlicz-Pettis theorem fails; namely weak-

L
p

{L_p}

sequence spaces

l

(

p
,
∞

)

l\left ( {p,\infty } \right )

for

0
>
p
≤
1

0 > p \leq 1

. We observe that if

0
>
p
>
1

0 > p > 1

then the space

l

(

p
,
∞

)

l\left ( {p,\infty } \right )

is nonseparable but separable with respect to its weak topology. Moreover, we show that the Orlicz-Pettis theorem holds for every Orlicz sequence space (even nonseparable).
Doubly-periodic solutions of a forced semilinear wave equation
Authors: Arias M. , Martínez-Amores P. , Ortega R.
Published: June 2010

Existence results are obtained for doubly-periodic solutions of a semilinear wave equation when the nonlinearity is bounded in one side.
Continuously homogeneous continua and their arc components
Authors: Prajs Janusz R.
Published: June 2010

Let

X
X

be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence

A
1

,

A
2

,
…

{A_1},{A_2}, \ldots

of its arc components with

X
=

c1

A
1

∪

c1

A
2

∪
⋯

X = {\text {c1}}{A_1} \cup {\text {c1}}{A_2} \cup \cdots

, and there is an arc component of

X
X

with nonempty interior, then

X
X

is arcwise connected. As consequences and applications we get: (1) if

X
X

is the countable union of arcwise connected continua, then

X
X

is arcwise connected; (2) if

X
X

is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.
Almost everywhere divergence of multiple Walsh-Fourier series
Authors: Harris David C.
Published: June 2010

C. Fefferman [1, 2, 3] has shown that the multiple Fourier series of an

f
∈

L
p

,
p
>
2

f \in {L^p},p > 2

, may diverge a.e. when summed over expanding spheres, but converges a.e. when summed over expanding polyhedral surfaces. We show this dichotomy does not prevail for multiple Walsh-Fourier series.