bluuebunny/crossref_metadata_2025 huggingface dataset:
Remarks on rings of constants of derivations
Authors: Li Wei
Published: June 2010

Let

k
k

be a field of characteristic

p
>
0

p > 0

and

D
≠
0

D \ne 0

a family of

k
k

-derivations of

k
[
x
,
y
]

k[x,y]

. We prove that

k

[
x
,
y

]
D

k{[x,y]^D}

,the ring of constants with respect to

D
D

, is a free

k
[

x
p

,

y
p

]

k[{x^p},{y^p}]

-module of rank

p
p

or 1 and

k

[
x
,
y

]
D

=
k
[

x
p

,

y
p

,

f
1

,
…
,

f

p
−
1

]

k{[x,y]^D} = k[{x^p},{y^p},{f_1}, \ldots ,{f_{p - 1}}]

for some

f
1

,
…
,

f

p
−
1

∈
k

[
x
,
y

]
D

{f_1}, \ldots ,{f_{p - 1}} \in k{[x,y]^D}

.
Uniqueness of aperiodic kneading sequences
Authors: Brucks K. M.
Published: June 2010

The trapezoidal function

f
e

(
x
)

{f_e}(x)

is defined for fixed

e
∈
(
0
,
1

/

2
)

e \in (0,1/2)

by

f
e

(
x
)
=
(
1

/

e
)
x

{f_e}(x) = (1/e)x

for

x
∈
[
0
,
e
]
,

f
e

(
x
)
=
1

x \in [0,e],{f_e}(x) = 1

for

x
∈
(
e
,
1
−
e
)

x \in (e,1 - e)

, and

f
e

(
x
)
=
(
1

/

e
)
(
1
−
x
)

{f_e}(x) = (1/e)(1 - x)

for

x
∈
[
1
−
e
,
1
]

x \in [1 - e,1]

. For a given

e
e

and the associated one-parameter family of maps

{
λ

f
e

(
x
)

|

λ
∈
[
0
,
1
]
}

\{ \lambda {f_e}(x)|\lambda \in [0,1]\}

, we show that if

A
A

is an aperiodic kneading sequence, then there is a unique

λ
∈
[
0
,
1
]

\lambda \in [0,1]

so that the itinerary of

λ
\lambda

under the map

λ

f
e

\lambda {f_e}

is

A
A

. From this, we conclude that the "stable windows" are dense in

[
0
,
1
]

[0,1]

for the one-parameter family

λ

f
e

\lambda {f_e}

.
On semisimple Mal′cev algebras
Authors: Elduque Alberto
Published: June 2010

Let

M
M

be a finite dimensional semisimple Malcev algebra over a perfect field of characteristic

≠
2
,
3

\ne 2,3

. Let

N
(
M
)

N(M)

be its

J
J

-nucleus and

J
(
M
,
M
,
M
)

J(M,M,M)

the subspace spanned by its jacobians. Then it is shown that

M
=
N
(
M
)
⊕
J
(
M
,
M
,
M
)
,
N
(
M
)

M = N(M) \oplus J(M,M,M),N(M)

is a semisimple Lie algebra and

J
(
M
,
M
,
M
)

J(M,M,M)

is a direct sum of simple non-Lie Malcev algebras.
A new example in 𝐾-theory of loopspaces
Authors: Ghazal Tahsin
Published: June 2010

The "Eilenberg-Moore" type spectral sequences which connect

K
∗

(
Ω
X
)

{K^ * }(\Omega X)

and

K
∗

(
X
)

{K^*}(X)

have well-known bad properties, when, for example,

X
=
K
(

Z

/

p
,
2
)

X = K(\mathbb {Z}/p,2)

. This paper shows that the result can be as bad when

X
X

is a finite complex.
Weighted weak-type (1,1) inequalities for rough operators
Authors: Hofmann Steve
Published: June 2010

Let

Ω
\Omega

be homogeneous of degree 0, have mean value 0 on the circle, and belong to

L
q

(

S
1

)

,
1
>
q
≤
∞

{L^q}\left ( {{S^1}} \right ),1 > q \leq \infty

. Then the two-dimensional operator defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T f left-parenthesis x right-parenthesis equals pv integral normal upper Omega left-parenthesis y right-parenthesis StartAbsoluteValue y EndAbsoluteValue Superscript negative 2 Baseline f left-parenthesis x minus y right-parenthesis d y">

T
f

(
x
)

=

pv

∫

Ω

(
y
)

|
y
|

−
2

f

(

x
−
y

)

d
y

Tf\left ( x \right ) = {\text {pv}}\int {\Omega \left ( y \right ){{\left | y \right |}^{ - 2}}} f\left ( {x - y} \right )dy

\]
is shown to be of weak-type

(
1
,
1
)

(1,1)

with respect to the weighted measures

|
x
|

α

d
x

{\left | x \right |^\alpha }dx

, if

−
2
+
1

/

q
>
α
>
0

- 2 + 1/q > \alpha > 0

. Under the weaker assumption that

Ω
\Omega

belongs to

L

log
+

L

(

S
1

)

L\,{\log ^ + }L\left ( {{S^1}} \right )

, the same result holds if

−
1
>
α
>
0

- 1 > \alpha > 0

. Similar results are also obtained for the related maximal operator
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript normal upper Omega Baseline f left-parenthesis x right-parenthesis equals sup Underscript r greater-than 0 Endscripts r Superscript negative 2 Baseline integral Underscript StartAbsoluteValue y EndAbsoluteValue greater-than r Endscripts StartAbsoluteValue normal upper Omega left-parenthesis y right-parenthesis f left-parenthesis x minus y right-parenthesis EndAbsoluteValue d y period">

M
Ω

f

(
x
)

=

sup

r
>
0

r

−
2

∫

|

y

|

>
r

|

Ω

(
y
)

f

(

x
−
y

)

|

d
y
.

{M_\Omega }f\left ( x \right ) = \sup \limits _{r > 0} {r^{ - 2}}\int _{|y| > r} {\left | {\Omega \left ( y \right )f\left ( {x - y} \right )} \right |dy.}

\]
Remarks on Chacon’s biting lemma
Authors: Ball J. M. , Murat F.
Published: June 2010

Chacon’s Biting Lemma states roughly that any bounded sequence in

L
1

{L^1}

possesses a subsequence converging weakly in

L
1

{L^1}

outside a decreasing family

E
k

{E_k}

of measurable sets with vanishingly small measure. A simple new proof of this result is presented that makes explicit which sets

E
k

{E_k}

need to be removed. The proof extends immediately to the case when the functions take values in a reflexive Banach space. The limit function is identified via the Young measure and approximations. The description of concentration provided by the lemma is discussed via a simple example.
Bonferroni-type inequalities via interpolating polynomials
Authors: Xu Yuan
Published: June 2010

Inequalities of Bonferroni-type are proved by means of the corresponding polynomial inequalities which are obtained from the Hermite interpolation.
Asymptotic behavior of solutions to Δ𝑢+𝐾𝑢^{𝜎}=0 on 𝑅ⁿ for 𝑛≥3
Authors: Trubek Jeanne
Published: June 2010

The equation

Δ
u
+
K

u
σ

=
0

\Delta u + K{u^\sigma } = 0

is considered in

R
n

{R^n}

for

n
≥
3
,
K

n \geq 3,K

a Hölder continuous function and

σ
\sigma

a positive constant. If

K
=
O
(

|
x
|

−
l

)

K = O({\left | x \right |^{ - l}})

for

l
>
2

l > 2

, we determine the asymptotic behavior of bounded solutions. In the case

K
K

is nonpositive and

σ
\sigma

is greater than one, we show that the first term in the asymptotic description may be chosen arbitrarily.
𝑂(2)×𝑂(2)-invariant hypersurfaces with constant negative scalar curvature in 𝐸⁴
Authors: Okayasu Takashi
Published: June 2010

We use the method of equivariant differential geometry to prove the existence of a complete hypersurface with constant negative scalar curvature in

E
n

(
n
≥
4
)

{E^n}(n \geq 4)

. This is the first example of a complete hypersurface with constant negative scalar curvature in

E
n

(
n
≥
4
)

{E^n}(n \geq 4)

.
Minimal harmonic functions on Denjoy domains
Authors: Gardiner Stephen J.
Published: June 2010

Let

Ω
=

R

n

∖
E

\Omega = {\mathbb {R}^n}\backslash E

, where

E
E

is a closed subset of the hyperplane

{

x
n

=
0

}

\left \{ {{x_n} = 0} \right \}

and every point of

E
E

is regular for the Dirichlet problem on

Ω
\Omega

. Further, let

α
k

{\alpha _k}

. denote the

(
n
−
1
)

(n - 1)

-dimensional measure of the set

{
X
∈
Ω
:

x
n

=
0
,

e
k

>

|

X

|

>

e

k
+
1

}

\{ X \in \Omega :{x_n} = 0,{e^k} > |X| > {e^{k + 1}}\}

. It is known that the cone,

P

E

{\mathcal {P}_E}

, of positive harmonic functions on

Ω
\Omega

which vanish on

E
E

has dimension 1 or 2. In this paper it is shown that if

∑

e

−
n
k

α
k

n

/

(
n
−
1
)

>
+
∞

\sum {{e^{ - nk}}\alpha _k^{n/(n - 1)} > + \infty }

then

dim
⁡

P

E

=
2

\dim {\mathcal {P}_E} = 2

. This result, which in the case

n
=
2

n = 2

implies a recent theorem of Segawa, is also shown to be sharp.
Convolution properties of a class of starlike functions
Authors: Singh Ram , Singh Sukhjit
Published: June 2010

Let

R
R

denote the class of functions

f
(
z
)
=
z
+

a
2

z
2

+
⋯

f(z) = z + {a_2}{z^2} + \cdots

that are analytic in the unit disc

E
=
{
z
:

|
z
|

>
1
}

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

and satisfy the condition

Re
⁡
(

f
′

(
z
)
+
z
f
(
z
)
)
>
0
,
z
∈
E

\operatorname {Re} (f’(z) + zf(z)) > 0,z \in E

. It is known that

R
R

is a subclass of

S
t

{S_t}

, the class of univalent starlike functions in

E
E

. In the present paper, among other things, we prove (i) for every

n
≥
1

n \geq 1

, the

n
n

th partial sum of

f
∈
R
,

s
n

(
z
,
f
)

f \in R,{s_n}(z,f)

, is univalent in

E
E

, (ii)

R
R

is closed with respect to Hadamard convolution, and (iii) the Hadamard convolution of any two members of

R
R

is a convex function in

E
E

.
Magnified curves on a flat torus, determination of almost periodic functions, and the Riemann-Lebesgue lemma
Authors: Strichartz Robert S.
Published: June 2010

Simple arguments, based on the Riemann-Lebesgue Lemma, are given to show that for a large class of curves

γ
\gamma

in

R

n

{{\mathbf {R}}^n}

, any almost periodic function is determined by its restriction to large dilates of

γ
\gamma

. Specializing to periodic functions, this means that magnified images of

γ
\gamma

on a flat torus tend to uniformly dense scribbles.
A weak-type estimate for Fourier-Bessel multipliers
Authors: Gosselin John , Stempak Krzysztof
Published: June 2010

We apply Hörmander’s technique to prove a weak-type

(
1
,
1
)

(1,1)

estimate for multiplier operators with respect to the Fourier-Bessel transform. This improves a result in [4, 5].
Pasting and modular lattices
Authors: Fried E. , Grätzer G.
Published: June 2010

A classical lattice construction of R. P. Dilworth is the gluing of two lattices. A number of recent papers by A. Slavík, A. Day, and J. Ježek investigated a generalization: pasting. In this note we prove that by pasting two finite modular lattices, one obtains a modular lattice.
Convergence of monotone dynamical systems with minimal equilibria
Authors: Wu Jian Hong
Published: June 2010

We show that each precompact orbit of strongly monotone dynamical systems on a Banach lattice

X
X

is convergent if there is a continuous map

e
:
X
→
E

e:X \to E

, the set of equilibria, such that

e
(
x
)

e(x)

is the maximal element in

E
E

with

e
(
x
)
≤
x

e(x) \leq x

. This result can be applied to study the convergence of a class of functional differential equations and partial differential equations.
On superposition of functions of bounded 𝜙-variation
Authors: Prus-Wiśniowski Franciszek
Published: June 2010

J. Ciemnoczolowski and W. Orlicz in [1] have obtained some results concerning superpositions of functions of bounded

φ
\varphi

-variation. In this note we show that the assumption in Theorem 1 of [1] that if

ψ
\psi

satisfies

Δ
2

{\Delta _2}

condition may be dropped. Moreover, Theorem 2.B of [1] is extended to a stronger version.
On Bloch’s constant
Authors: Bonk Mario
Published: June 2010

The lower bound for Bloch’s constant is slightly improved.
A nonstandard delta function
Authors: Todorov Todor
Published: June 2010

We prove that the Dirac delta distribution has a kernel in the class of the pointwise nonstandard functions.
Weak amenability of group algebras of connected complex semisimple Lie groups
Authors: Johnson B. E.
Published: June 2010

We consider the problem of whether every continuous derivation from a group algebra

L
1

(
G
)

{L^1}(G)

into its dual

L
∞

(
G
)

{L^\infty }(G)

(where the

L
1

(
G
)

{L^1}(G)

actions on

L
∞

(
G
)

{L^\infty }(G)

are the adjoint of multiplication in

L
1

(
G
)

{L^1}(G)

is inner, that is, of the form

D
(
a
)
=
a
F
−
F
a

D(a) = aF - Fa

for some

F
∈

L
∞

(
G
)

F \in {L^\infty }(G)

. This had previously been established to hold for discrete and amenable groups and is now established for

G
=

Gl(

n
,

C

)

G = {\text {Gl(}}n,{\mathbf {C}})

and for all connected semisimple complex Lie groups.
Holomorphic motions and polynomial hulls
Authors: Slodkowski Zbigniew
Published: June 2010

A holomorphic motion of

E
⊂

C

E \subset \mathbb {C}

over the unit disc

D
D

is a map

f
:
D
×

C

→

C

f:D \times \mathbb {C} \to \mathbb {C}

such that

f
(
0
,
w
)
=
w
,
w
∈
E

f(0,w) = w,w \in E

, the function

f
(
z
,
w
)
=

f
z

(
w
)

f(z,w) = {f_z}(w)

is holomorphic in

z
z

, and

f
z

:
E
→

C

{f_z}:E \to \mathbb {C}

is an injection for all

z
∈
D

z \in D

. Answering a question posed by Sullivan and Thurston [13], we show that every such

f
f

can be extended to a holomorphic motion

F
:
D
×

C

→

C

F:D \times \mathbb {C} \to \mathbb {C}

. As a main step a "holomorphic axiom of choice" is obtained (concerning selections from the sets

C

∖

f
z

(
E
)
,
z
∈
D
)

\mathbb {C}\backslash {f_z}(E),z \in D)

. The proof uses earlier results on the existence of analytic discs in the polynomial hulls of some subsets of

C

2

{\mathbb {C}^2}

.
Matrices with circular symmetry on their unitary orbits and 𝐶-numerical ranges
Authors: Li Chi-Kwong , Tsing Nam-Kiu
Published: June 2010

We give equivalent characterizations for those

n
×
n

n \times n

complex matrices

A
A

whose unitary orbits

U

(
A
)

\mathcal {U}(A)

and

C
C

-numerical ranges

W
C

(
A
)

{W_C}(A)

satisfy

e

i
θ

U

(
A
)
=

U

(
A
)

{e^{i\theta }}\mathcal {U}(A) = \mathcal {U}(A)

or

e

i
θ

W
C

(
A
)
=

W
C

(
A
)

{e^{i\theta }}{W_C}(A) = {W_C}(A)

for some real

θ
\theta

(or for all real

θ
\theta

). In particular, we show that they are the block-cyclic or block-shift operators. Some of these results are extended to infinite-dimensional Hilbert spaces.
Free subalgebras of enveloping fields
Authors: Makar-Limanov L. , Malcolmson P.
Published: June 2010

Let

K
K

be a field of characteristic zero and

L
L

a nonabelian Lie algebra over

K
K

. We show that the skew field of fractions of the enveloping algebra of

L
L

over

K
K

contains a free noncommutative

K
K

-algebra when

L
L

is solvable or finite-dimensional.
On rings for which homogeneous maps are linear
Authors: Fuchs P. , Maxson C. J. , Pilz G.
Published: June 2010

Let

R
R

be the collection of all rings

R
R

such that for every

R
R

-module

G
G

, the centralizer near-ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript upper R Baseline left-parenthesis upper G right-parenthesis equals left-brace f colon upper G right-arrow upper G vertical-bar f left-parenthesis r x right-parenthesis equals r f left-parenthesis x right-parenthesis comma r element-of upper R comma x element-of upper G right-brace">

M
R

(
G
)
=
{
f
:
G
→
G

|

f
(
r
x
)
=
r
f
(
x
)
,
r
∈
R
,
x
∈
G
}

{M_R}(G) = \{ f:G \to G|f(rx) = rf(x),r \in R,x \in G\}

is a ring. We show

R
∈
R

R \in R

if and only if

M
R

(
G
)
=

En

d

R

(
G
)

{M_R}(G) = {\text {En}}{{\text {d}}_R}(G)

for each

R
R

-module

G
G

. Further information about

R
R

is collected and the Artinian rings in

R
R

are completely characterized.
A class of absolute retracts in spaces of integrable functions
Authors: Bressan Alberto , Cellina Arrigo , Fryszkowski Andrzej
Published: June 2010

We consider a class of subsets of

L
1

{L^1}

, that are shown to be absolute retracts, that contains at once decomposable sets and sets of solutions to Lipschitzean differential inclusions. In this way we generalize and unify a number of different previous results.
Variation of parameters formula for the equation of Cooke and Wiener
Authors: Jayasree K. N. , Deo S. G.
Published: June 2010

A variation of parameters and Gronwall type integral inequality are proved for a differential equation involving piecewise alternately retarded and advanced argument.
On solutions of linear differential equations with real zeros; proof of a conjecture of Hellerstein and Rossi
Authors: Brüggemann Franz
Published: June 2010

We prove the following conjecture that is due to Hellerstein and Rossi: Let

{

w
1

,
…
,

w
n

}

\left \{ {{w_1}, \ldots ,{w_n}} \right \}

be a fundamental system of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L w equals w Superscript left-parenthesis n right-parenthesis Baseline plus a Subscript n minus 1 Baseline left-parenthesis z right-parenthesis w Superscript left-parenthesis n minus 1 right-parenthesis Baseline plus midline-horizontal-ellipsis plus a 0 left-parenthesis z right-parenthesis w identical-to 0">

L
w
=

w

(
n
)

+

a

n
−
1

(
z
)

w

(
n
−
1
)

+
⋯
+

a
0

(
z
)
w
≡
0

Lw = {w^{(n)}} + {a_{n - 1}}(z){w^{(n - 1)}} + \cdots + {a_0}(z)w \equiv 0

\]
with polynomials

a
j

(
z
)
(
0
≤
j
≤
n
−
1
)

{a_j}(z)(0 \leq j \leq n - 1)

. If each

w
k

(
1
≤
k
≤
n
)

{w_k}(1 \leq k \leq n)

has only finitely many nonreal zeros, then there exists a polynomial

q
(
z
)

q(z)

such that

u
k

:=
exp
⁡
(
q
(
z
)
)

w
k

(
1
≤
k
≤
n
)

{u_k}: = \exp (q(z)){w_k}(1 \leq k \leq n)

form a fundamental system of a homogeneous linear differential equation with constant coefficients.
Spectral properties of perturbed linear operators and their application to infinite matrices
Authors: Farid F. O.
Published: June 2010

In this paper, upper bounds for the difference between the eigenvalues and the eigenvectors of a closed linear operator

D
D

and those of

D
+
F

D + F

, where

F
F

is a bounded linear operator, are given in terms of the norm of

F
F

. These results are applied to approximate the eigenvalues and the eigenvectors of a diagonally infinite matrix by those of its corresponding diagonal matrix.
On the rank and the crank modulo 4
Authors: Lewis Richard
Published: June 2010

Let

N
(
r
,
m
,
n
)

N(r,m,n)

(respectively,

M
(
r
,
m
,
n
)

M(r,m,n)

) denote the number of partitions of

n
n

whose ranks (respectively, cranks) are congruent to

r
r

modulo

n
n

. It is shown that

N
(
0
,
4
,
2
n
+
1
)
=
M
(
1
,
4
,
2
n
+
1
)

N(0,4,2n + 1) = M(1,4,2n + 1)

and

N
(
2
,
4
,
2
n
)
=
M
(
1
,
4
,
2
n
)

N(2,4,2n) = M(1,4,2n)

.
Positive sequence topological entropy characterizes chaotic maps
Authors: Franzová N. , Smítal J.
Published: June 2010

We prove that a continuous map

f
f

of the interval is chaotic (in the sense of Li and Yorke) iff its sequence topological entropy

h
A

(
f
)

{h_A}(f)

relative to a suitable increasing sequence

A
A

of times is positive. This result is interesting since the ordinary topological entropy

h
(
f
)

h(f)

of chaotic maps can be zero.
Some operator-theoretic calculus for positive definite kernels
Authors: Athavale Ameer
Published: June 2010

If

κ
\kappa

is a positive definite kernel on the open unit disk

D
D

in the complex plane, then we associate with it a positive definite kernel

κ
′

\kappa ’

on

D
D

and correlate some operator theoretic properties of

M

(
κ
)

M\left ( \kappa \right )

and

M

(

κ
′

)

M\left ( {\kappa ’} \right )

, where

M

(
κ
)

M\left ( \kappa \right )

denotes the multiplication operator on the functional Hilbert space

H

(
κ
)

\mathcal {H}\left ( \kappa \right )

associated with

κ
\kappa

. The main emphasis of this paper is on the discussion of hyponormality and subnormality properties. We also construct a sequence of positive definite kernels

κ

−
p

(

p
=
1
,
2
,
…

)

{\kappa _{ - p}}\left ( {p = 1,2, \ldots } \right )

on

D
D

such that

M

(

κ

−
p

)

M\left ( {{\kappa _{ - p}}} \right )

is a

(

p
+
1

)

\left ( {p + 1} \right )

-isometry, but not a

q
q

-isometry for any positive integer

q
q

less than or equal to

p
p

.
Two examples of Borel partially ordered sets with the countable chain condition
Authors: Todorčević Stevo
Published: June 2010

We define an open symmetric two-place relation on the reals such that the reals cannot be covered by countably many sets of related elements, but there is no uncountable set of unrelated elements. The poset

P

\mathcal {P}

of finite sets of related elements satisfies the countable chain condition but it may fail to have the property K, i.e., a substantial irregularity can be injected in

P

\mathcal {P}

.
A function which is arc-analytic but not continuous
Authors: Bierstone E. , Milman P. D. , Parusiński A.
Published: June 2010

We construct an arc-analytic function (i.e., a function analytic on every real analytic arc) which is not continuous, as well as a continuous arc-analytic function which is not subanalytic.
A note on the connectedness problem for nest algebras
Authors: Pitts David R.
Published: June 2010

It has been conjectured that a certain operator

T
T

belonging to the group

G

\mathcal {G}

of invertible elements of the algebra

Alg
⁡

Z

\operatorname {Alg} {\mathbf {Z}}

of doubly infinite upper-triangular bounded matrices lies outside the connected component of the identity in

G

\mathcal {G}

. In this note we show that

T
T

actually lies inside the connected component of the identity of

G

\mathcal {G}

.
Generalized hypergeometric functions at unit argument
Authors: Bühring Wolfgang
Published: June 2010

The analytic continuation near

z
=
1

z = 1

of the hypergeometric function

p
+
1

F
p

(
z
)

_{p + 1}{F_p}\left ( z \right )

is obtained for arbitrary

p
=
2
,
3
,
…
,

p = 2,3, \ldots ,

, including the exceptional cases when the sum of the denominator parameters minus the sum of the numerator parameters is equal to an integer.
A geometric interpretation of Segal’s inequality ‖𝑒^{𝑋+𝑌}‖≤‖𝑒^{𝑋/2}𝑒^{𝑌}𝑒^{𝑋/2}‖
Authors: Corach G. , Porta H. , Recht L.
Published: June 2010

It is shown that the exponential mapping of the manifold of positive elements of a

C
∗

{C^*}

-algebra (provided with its natural connection) increases distances (when measured in the natural Finsler structure). The proof relies on Segal’s inequality
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue e Superscript upper X plus upper Y Baseline StartAbsoluteValue EndAbsoluteValue less-than-or-equal-to StartAbsoluteValue EndAbsoluteValue e Superscript upper X slash 2 Baseline e Superscript upper Y Baseline e Superscript upper X slash 2 Baseline StartAbsoluteValue EndAbsoluteValue">

|

|

e

X
+
Y

|

|

≤

|

|

e

X

/

2

e
Y

e

X

/

2

|

|

||{e^{X + Y}}|| \leq ||{e^{X/2}}{e^Y}{e^{X/2}}||

, valid for all symmetric

X
,
Y

X,Y

in any

C
∗

{C^*}

-algebra. In turn, this geometric inequality implies Segal’s’ inequality.
A result concerning derivations in Banach algebras
Authors: Vukman J.
Published: June 2010

The main result: Let

A
A

be a Banach algebra over the complex field

C
C

. Suppose there exists a continuous derivation

D
:
A
→
A

D:A \to A

, such that

α

D
3

+

D
2

\alpha {D^3} + {D^2}

is a derivation for some

α
∈
C

\alpha \in C

. In this case

D
D

maps

A
A

into its radical.
Interpolating operators in nest algebras
Authors: Anoussis M.
Published: June 2010

Given two families

{

x
α

}
,
{

y
α

}

\{ {x_\alpha }\} ,\{ {y_\alpha }\}

of vectors and a nest

N
N

in a Hilbert space

H
H

, we provide a necessary and sufficient condition for the existence of an operator

T
T

in the nest algebra satisfying

T

x
α

=

y
α

T{x_\alpha } = {y_\alpha }

for every

α
\alpha

.
Smooth approximations in Banach spaces
Authors: Vanderwerff J.
Published: June 2010

A Banach space that has a locally uniformly convex (LUC) norm whose dual is also LUC is shown to admit

C
1

{C^1}

-smooth partitions of unity. It is also established that there is a norm on a Hubert space with Lipschitz derivative that cannot be approximated uniformly on bounded sets by functions with uniformly continuous second derivative.
Shadowing property of continuous maps
Authors: Gedeon Tomáš , Kuchta Milan
Published: June 2010

We study continuous maps of an interval into itself. We find the necessary and sufficient condition for the maps of the type

2
n

{2^n}

to have the shadowing property. Further we show that any chaotic map, which has only cycles of order a power of 2, does not have the shadowing property.
Oscillation criteria for second order nonlinear differential equations with integrable coefficients
Authors: Wong James S. W.
Published: June 2010

Consider the second order nonlinear differential equation

y
+
a

(
t
)

f

(
y
)

=
0

y + a\left ( t \right )f\left ( y \right ) = 0

where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a left-parenthesis t right-parenthesis element-of upper C left-bracket 0 comma normal infinity right-parenthesis comma f left-parenthesis y right-parenthesis element-of upper C Superscript 1 Baseline left-parenthesis negative normal infinity comma normal infinity right-parenthesis comma f prime left-parenthesis y right-parenthesis greater-than-or-equal-to 0">

a

(
t
)

∈
C
[
0
,
∞
)
,
f

(
y
)

∈

C
1

(

−
∞
,
∞

)

,

f
′

(
y
)

≥
0

a\left ( t \right ) \in C[0,\infty ),f\left ( y \right ) \in {C^1}\left ( { - \infty ,\infty } \right ),f’\left ( y \right ) \geq 0

, and

y
f

(
y
)

>
0

yf\left ( y \right ) > 0

for

y

≠

0

{\text {y}} \ne {\text {0}}

. Furthermore,

f

(
y
)

f\left ( y \right )

also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function

f

(
y
)

=

|

y

|

γ

sgn
⁡
y

f\left ( y \right ) = |y{|^\gamma }\operatorname {sgn} y

with

γ

 > 1

\gamma {\text { > 1}}

and

0
>
γ
>
1

0 > \gamma > 1

respectively. The coefficient

a

(
t
)

a\left ( t \right )

is allowed to be negative for arbitrarily large values of

t
t

and is integrable in the sense that the improper interval

∫
t
∞

a

(
s
)

d
s

=
A

(
t
)

\int _t^\infty {a\left ( s \right )ds} = A\left ( t \right )

exists for each

t
≥
0

t \geq 0

. Oscillation criteria involving integrals of

A

(
t
)

A\left ( t \right )

due to Coles and Butler for the superlinear and sublinear cases are shown to remain valid without the additional hypothesis that

A

(
t
)

≥
0

A\left ( t \right ) \geq 0

.
An invariant on 3-dimensional Lie algebras
Authors: Tasaki Hiroyuki , Umehara Masaaki
Published: June 2010

We construct an extra symmetric bilinear form on a

3
3

-dimensional Lie algebra

g

\mathfrak {g}

which induces an invariant

χ

(

g

)

\chi \left ( \mathfrak {g} \right )

on

g

\mathfrak {g}

. Moreover it provides a new viewpoint for the classical classification of

3
3

-dimensional Lie algebras
Random walk in a Weyl chamber
Authors: Gessel Ira M. , Zeilberger Doron
Published: June 2010

The classical Ballot problem that counts the number of ways of walking from the origin and staying within the wedge

x
1

≥

x
2

≥
⋯
≥

x
n

{x_1} \geq {x_2} \geq \cdots \geq {x_n}

(which is a Weyl chamber for the symmetric group), using positive unit steps, is generalized to general Weyl groups and general sets of steps.
On the number of solutions of the equation 𝑥^{𝑝^{𝑘}}=𝑎 in a finite 𝑝-group
Authors: Berkovich Yakov G.
Published: June 2010

A. Kulakoff (Math. Ann. 104 (1931), 778-793) proved that for

p
>
2

p > 2

the number of solutions of the equation

x

p
k

=
e

{x^{{p^k}}} = e

(

e
e

is a unit element of

G
G

) in a finite noncyclic

p
p

-group

G
G

is divisible by

p

k
+
1

{p^{k + 1}}

if

exp
⁡
G
≥

p
k

\operatorname {exp} G \geq {p^k}

. In this note we consider the number

N
(
a
,
G
,
k
)

N(a,G,k)

of solutions of the equation

x

p
k

=
a

{x^{{p^k}}} = a

in

G
,

a
∈
G

G,\;a \in G

. Our results cover the case

p
=
2

p = 2

also.
Linear combinations of projections in von Neumann algebras
Authors: Goldstein Stanisław , Paszkiewicz Adam
Published: June 2010

Any operator in a von Neumann algebra is a linear combination of a finite number of projections from the algebra with coefficients from the center of the algebra. Those von Neumann algebras that are the complex linear span of their projections are identified.
Shadowing property for inverse limit spaces
Authors: Chen Liang , Li Shi Hai
Published: June 2010

We investigate the relationship between the shadowing property for continuous maps on a compact metric space and that for the shift maps on the inverse limit spaces. As an example, we show that the shift map of some pseudoarc has the shadowing property.
Kantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces
Authors: Hanin Leonid G.
Published: June 2010

We obtain necessary and sufficient conditions on a compact metric space

(

K
,
ρ

)

\left ( {K,\rho } \right )

that provide a natural isometric isomorphism between completion of the space of Borel measures on

K
K

with the Kantorovich-Rubinstein norm and the space

(

lip
⁡
(

K
,
ρ

)

)
∗

{( {\operatorname {lip}( {K,\rho } )} )^*}

or equivalently between the spaces

Lip
⁡
(

K
,
ρ

)

\operatorname {Lip}( {K,\rho } )

and

(

lip
⁡
(

K
,
ρ

)

)

∗
∗

{( {\operatorname {lip}( {K,\rho } )} )^{**}}

. Such metric spaces are studied and related properties of Lipschitz spaces are established.
Regular and purely irregular bounded charges: a decomposition theorem
Authors: Girotto Bruno , Holzer Silvano
Published: June 2010

We introduce the notions of regular and purely irregular charges with respect to a pair of pavings and study their structural properties. Moreover, we link regularity and

σ
\sigma

-additivity, obtaining some generalizations of well-known theorems. Finally, when the pavings satisfy some reasonable weak conditions, we can decompose any bounded charge into regular and purely irregular decomposants; this decomposition becomes the Hewitt-Yosida one, whenever the charges are defined on the Baire

σ
\sigma

-field of a countably compact space.
A remark on Sakai’s quadratic Radon-Nikodým theorem
Authors: Kosaki Hideki
Published: June 2010

Sakai’s Radon-Nikodym theorem (in a quadratic form) for normal states on a von Neumann algebra is considered. We show that the conclusion of this theorem follows from a much weaker order assumption on involved states.
Interpolating functions of minimal norm, star-invariant subspaces and kernels of Toeplitz operators
Authors: Dyakonov Konstantin M.
Published: June 2010

It is proved that for each inner function

θ
\theta

there exists an interpolating sequence

{

z
n

}

\left \{ {{z_n}} \right \}

in the disk such that

sup
n

|

θ
(

z
n

)

|

>
1

{\sup _n}|\theta ({z_n})| > 1

, but every function

g
g

in

H
∞

{H^\infty }

with

g
(

z
n

)
=
θ
(

z
n

)
(
n
=
1
,
2
,
…
)

g({z_n}) = \theta ({z_n})(n = 1,2, \ldots )

satisfies

|

|

g

|

|

∞

≥
1

||g|{|_\infty } \geq 1

. Some results are obtained concerning interpolation in the star-invariant subspace

H
2

⊖
θ

H
2

{H^2} \ominus \theta {H^2}

. This paper also contains a "geometric" result connected with kernels of Toeplitz operators.
On a problem of Nirenberg concerning expanding maps in Hilbert space
Authors: Szczepański Janusz
Published: June 2010

Let

H

{\mathbf {H}}

be a Hilbert space and

f
:

H

→

H

f:{\mathbf {H}} \to {\mathbf {H}}

a continuous map which is expanding (i.e.,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue f left-parenthesis bold x right-parenthesis minus f left-parenthesis bold y right-parenthesis StartAbsoluteValue EndAbsoluteValue greater-than-or-equal-to StartAbsoluteValue EndAbsoluteValue bold x minus bold y StartAbsoluteValue EndAbsoluteValue">

|

|

f
(

x

)
−
f
(

y

)

|

|

≥

|

|

x

−

y

|

|

||f({\mathbf {x}}) - f({\mathbf {y}})|| \geq ||{\mathbf {x}} - {\mathbf {y}}||

for all

x

,

y

∈

H

{\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}

) and such that

f
(

H

)

f({\mathbf {H}})

has nonempty interior. Are these conditions sufficient to ensure that

f
f

is onto? This question was stated by Nirenberg in 1974. In this paper we give a partial negative answer to this problem; namely, we present an example of a map

F
:

H

→

H

F:{\mathbf {H}} \to {\mathbf {H}}

which is not onto, continuous,

F
(

H

)

F({\mathbf {H}})

has nonempty interior, and for every

x

,

y

∈

H

{\mathbf {x}},{\mathbf {y}} \in {\mathbf {H}}

there is

n
0

∈

N

{n_0} \in \mathbb {N}

(depending on

x

{\mathbf {x}}

and

y

{\mathbf {y}}

) such that for every

n
≥

n
0

n \geq {n_0}

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue upper F Superscript n Baseline left-parenthesis bold x right-parenthesis minus upper F Superscript n Baseline left-parenthesis bold y right-parenthesis StartAbsoluteValue EndAbsoluteValue greater-than-or-equal-to c Superscript n minus m Baseline StartAbsoluteValue EndAbsoluteValue bold x minus bold y StartAbsoluteValue EndAbsoluteValue">

|

|

F
n

(

x

)
−

F
n

(

y

)

|

|

≥

c

n
−
m

|

|

x

−

y

|

|

||{F^n}({\mathbf {x}}) - {F^n}({\mathbf {y}})|| \geq {c^{n - m}}||{\mathbf {x}} - {\mathbf {y}}||

\]
where

F
n

{F^n}

is the

n
n

th iterate of the map

F
,
c

F,c

is a constant greater than 2, and

m
m

is an integer depending on

x

{\mathbf {x}}

and

y

{\mathbf {y}}

. Our example satisfies

|

|

F
(

x

)

|

|

=
c

|

|

x

|

|

||F({\mathbf {x}})|| = c||{\mathbf {x}}||

for all

x

∈

H

{\mathbf {x}} \in {\mathbf {H}}

. We show that no map with the above properties exists in the finite-dimensional case.
A characterization of inner ideals in 𝐽𝐵*-triples
Authors: Edwards C. M. , Rüttimann G. T.
Published: June 2010

It is shown that a norm-closed subtriple

B
B

of a

J

B
∗

J{B^ * }

-triple

A
A

is an inner ideal if and only if every bounded linear functional on

B
B

has a unique norm-preserving extension to a bounded linear functional on

A
A

. It follows that the norm-closed subtriples

B
B

of a

C
∗

{C^ * }

-algebra

A
A

that enjoy this unique extension property are precisely those of the form

e

A

∗
∗

f
∩
A

e{A^{ * * }}f \cap A

where

(
e
,
f
)

(e,f)

is a pair of centrally equivalent open projections in the

W
∗

{W^ * }

-envelope

A

∗
∗

{A^{ * * }}

of

A
A

.
The dual dynamic programming
Authors: Nowakowski Andrzej
Published: June 2010

The dual approach to dynamic programming for the generalized problem of Bolza is described. A suitable verification theorem is proved and a dual optimal feedback control is introduced.
The 𝑝-periodicity of the mapping class group and the estimate of its 𝑝-period
Authors: Xia Yining
Published: June 2010

We determine completely the primes

p
p

for which the Farrell-Tate cohomology of the mapping class group

Γ
g

{\Gamma _g}

is

p
p

-periodic. We also estimate the

p
p

-period of a

p
p

-periodic

Γ
g

{\Gamma _g}

.
Commutator ideals and semicommutator ideals of Toeplitz algebras associated with flows
Authors: Muhly Paul S. , Xia Jingbo
Published: June 2010

We prove that for a Toeplitz

C
∗

{C^ * }

-algebra associated with a flow that has no fixed points, the commutator ideal and the semicommutator ideal coincide.
Asymptotic behavior of a linear delay difference equation
Authors: Driver R. D. , Ladas G. , Vlahos P. N.
Published: June 2010

Consider the linear delay difference equations\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript n plus 1 Baseline minus x Subscript n Baseline equals sigma-summation Underscript j equals 1 Overscript m Endscripts a Subscript j Baseline left-parenthesis x Subscript n minus k Sub Subscript j Subscript Baseline minus x Subscript n minus l Sub Subscript j Subscript Baseline right-parenthesis comma n equals 0 comma 1 comma 2 comma ellipsis">xn+1−xn=∑j=1maj(xn−kj−xn−lj),n=0,1,2,…{x_{n + 1}} - {x_n} = \sum \limits _{j = 1}^m {{a_j}({x_{n - {k_j}}} - {x_{n - {l_j}}}),\quad n = 0,1,2, \ldots }\]and\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y Subscript n plus 1 Baseline minus y Subscript n Baseline equals sigma-summation Underscript j equals 1 Overscript k Endscripts b Subscript j Baseline y Subscript n minus j Baseline comma n equals 0 comma 1 comma 2 comma ellipsis comma">yn+1−yn=∑j=1kbjyn−j,n=0,1,2,…,{y_{n + 1}} - {y_n} = \sum \limits _{j = 1}^k {{b_j}{y_{n - j}},\quad n = 0,1,2, \ldots ,}\]where the coefficientsaj{a_j}andbj{b_j}are real andkj{k_j}andlj{l_j}are nonnegative integers. In this note we describe, in terms of the initial conditions, the asymptotic behavior of solutions of these equations in several cases when the characteristic equation has a dominant real root. Some of the results extend to systems of equations.
Continuous functions on the interval which have conjugates near the identity
Authors: Young Sam W.
Published: June 2010

This paper concerns continuous functions on the unit interval and provides solutions to two problems: Characterize those continuous functions that are topologically equivalent (resp. conjugate) to functions arbitrarily near the identity. The second question was raised by Joe Martin.
A characterization of epi-convergence in terms of convergence of level sets
Authors: Beer Gerald , Rockafellar R. T. , Wets Roger J.-B.
Published: June 2010

Let

LSC
⁡
(
X
)

\operatorname {LSC} (X)

denote the extended real-valued lower semicontinuous functions on a separable metrizable space

X
X

. We show that a sequence

⟨

f
n

⟩

\left \langle {{f_n}} \right \rangle

in

LSC
⁡
(
X
)

\operatorname {LSC} (X)

is epi-convergent to

f
∈
LSC
⁡
(
X
)

f \in \operatorname {LSC} (X)

if and only for each real

α
\alpha

, the level set of height

α
\alpha

of

f
f

can be recovered as the Painlevé-Kuratowski limit of an appropriately chosen sequence of level sets of the

f
n

{f_n}

at heights

α
n

{\alpha _n}

approaching

α
\alpha

. Assuming the continuum hypothesis, this result fails without separability. An analogous result holds for weakly lower semicontinuous functions defined on a separable Banach space with respect to Mosco epi-convergence.
Nonexistence of measurable optimal selections
Authors: Burgess John , Maitra Ashok
Published: June 2010

We give an example of a function

f
f

on a separable metric space

X
X

into a compact metric space

Y
Y

such that the graph of

f
f

is a Borel subset of

X
×
Y

X \times Y

, but

f
f

is not Borel measurable. The example forms the basis for our construction of an upper semicontinuous, compact model of a one-day dynamic programming problem where the player has an optimal action at each state, but is unable to make a choice of such an action in a Borel measurable manner.
Rigidity theorems for nonpositive Einstein metrics
Authors: Shen Zhong Min
Published: June 2010

In this paper we study the following problem: When must a complete Einstein metric

g
g

on an

n
n

-manifold with

Ric
=
(
n
−
1
)
λ

g,

λ
≤
0

\operatorname {Ric} = (n - 1)\lambda {\text {g,}}\lambda \leq 0

, be a metric of constant curvature

λ
?

\lambda ?
Least area tori in 3-manifolds
Authors: Scott G. P. , Swarup G. A.
Published: June 2010

We consider incompressible maps of the torus into a

3
3

-manifold that have least possible area among all such maps. We show that such a map must be an embedding in many cases.
The long-time behavior of geodesics in certain left-invariant metrics
Authors: D’Atri J. E.
Published: June 2010

We introduce a new notion to describe the behavior of geodesics as the parameter increases to infinity and obtain quantitative results for a class of left-invariant metrics that includes the Bergman metric on bounded homogeneous domains.
Level sets of density continuous functions
Authors: Ciesielski Krzysztof , Larson Lee
Published: June 2010

A function

f
:

R

→

R

f:{\mathbf {R}} \to {\mathbf {R}}

is density continuous if it is continuous when both its range and domain are endowed with the density topology. The level sets of density continuous functions are characterised as those sets which are density closed and ambiguous.
Le Canada semi-dry
Authors: Loveys James G. , Wagner Frank O.
Published: June 2010

We show that a simple group

G
G

of finite Morley Rank acting faithfully on as divisible abelian group must be a linear group over some algebraically closed field.
Irregularities in the distribution of irreducible polynomials
Authors: McCurley Kevin S.
Published: June 2010

We prove that there exist monic polynomials

f
f

over

GF
⁡
(
q
)

\operatorname {GF} (q)

for which

f
+
g

f + g

is reducible for all

g
∈
GF
⁡
(
q
)
[
x
]

g \in \operatorname {GF} (q)[x]

with small degree. This is the analogue for polynomials of a result of Erdös and Rankin concerning gaps between consecutive primes.
Topological completions of metrizable spaces
Authors: Fitzpatrick Ben , Gruenhage Gary F. , Ott James W.
Published: June 2010

For a pair of metrizable spaces

X
X

and

Y
Y

, we investigate conditions under which there is a dense embedding

h
:
X
→
Z

h:X \to Z

, where

Z
Z

is completely metrizable and

Z
∖
h
(
X
)

Z\backslash h(X)

is homeomorphic to

Y
Y

. In such a case,

Z
Z

is called a topological completion of

X
X

and

Y
Y

is called a completion remainder of

X
X

. In case

X
X

and

Y
Y

are completely metrizable, we give necessary and sufficient conditions that

Y
Y

be a completion remainder of

X
X

. We characterize the completion remainders of

R

{\mathbf {R}}

and those of the rationals,

Q

{\mathbf {Q}}

. We also characterize the remainders of

Q

(
κ
)

{\mathbf {Q}}(\kappa )

, a nonseparable analogue of

Q

{\mathbf {Q}}

.
The number of solutions of a certain quadratic congruence related to the class number of 𝑄(√𝑝)
Authors: Le Mao Hua
Published: June 2010

Let

p
p

be an odd prime, and let

k
k

be a positive integer with

i
⩽
k
⩽
(
p
−
1
)

/

2

i \leqslant k \leqslant (p - 1)/2

. In this note we give a formula for the number of solutions

(

x
1

,
…
,

x
k

)

({x_1}, \ldots ,{x_k})

of the congruence

x
1
2

+
⋯
+

x
k
2

≡
0

(
mod
p
)

x_1^2 + \cdots + x_k^2 \equiv 0\;(\bmod p)

,

1
⩽

x
1

>
⋯
>

x
k

⩽
(
p
−
1
)

/

2

1 \leqslant {x_1} > \cdots > {x_k} \leqslant (p - 1)/2

.
Toward a precise smoothness hypothesis in Sard’s theorem
Authors: Bates S. M.
Published: June 2010

The familiar

C

k

{{\mathbf {C}}^k}

smoothness hypothesis in the Morse-Sard Theorem can be weakened to

C

k
−
1
,
1

{{\mathbf {C}}^{k - 1,1}}

.
Refinements of Ky Fan’s inequality
Authors: Alzer Horst
Published: June 2010

We prove the inequalities
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A prime Subscript n Baseline slash upper G Subscript n Superscript prime Baseline less-than-or-slanted-equals left-parenthesis 1 minus upper G prime Subscript n right-parenthesis slash left-parenthesis 1 minus upper A prime Subscript n right-parenthesis less-than-or-slanted-equals upper A Subscript n Baseline slash upper G Subscript n">

A
n
′

/

G
n
′

⩽
(
1
−

G
n
′

)

/

(
1
−

A
n
′

)
⩽

A
n

/

G
n

A_n’ /G_n’ \leqslant (1 - G_n’ )/(1 - A_n’ ) \leqslant {A_n}/{G_n}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A prime Subscript n Baseline slash upper G Subscript n Superscript prime Baseline less-than-or-slanted-equals left-parenthesis 1 minus upper G Subscript n Baseline right-parenthesis slash left-parenthesis 1 minus upper A Subscript n Baseline right-parenthesis less-than-or-slanted-equals upper A Subscript n Baseline slash upper G Subscript n Baseline comma">

A
n
′

/

G
n
′

⩽
(
1
−

G
n

)

/

(
1
−

A
n

)
⩽

A
n

/

G
n

,

A_n’ /G_n’ \leqslant (1 - {G_n})/(1 - {A_n}) \leqslant {A_n}/{G_n},

\]
where

A
n

{A_n}

and

G
n

{G_n}

(respectively,

A
n
′

A_n’

and

G
n
′

G_n’

) denote the unweighted arithmetic and geometric means of

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

(respectively,

1
−

x
1

,
…
,

1
−

x
n

1 - {x_1}, \ldots ,\;1 - {x_n}

) with

x
i

∈
(
0
,

1
2

]
(
i
=
1
,
…
,
n
;
n
⩾
2

{x_i} \in (0,\tfrac {1} {2}](i = 1, \ldots ,n;n \geqslant 2

. Further we show that the ratios

(
1
−

G
n
′

)

/

(
1
−

A
n
′

)

(1 - G_n’ )/(1 - A_n’)

and

(
1
−

G
n

)

/

(
1
−

A
n

)

(1 - {G_n})/(1 - {A_n})

can be compared if and only if

n
=
2

n = 2

.
𝐴₂-annihilated elements in 𝐻_{*}(ΩΣ𝑅𝑃²)
Authors: Anick D. J. , Peterson F. P.
Published: June 2010

Let

X
n

{X^n}

denote the smash product of

n
n

copies of

R

P

2

\mathbb {R}{\mathbb {P}^2}

. We describe a minimal set of generators for

H

∗

(

X
n

;

Z

2

)

{H^{\ast }}({X^n};{\mathbb {Z}_2})

as a module over the

mod

2

\bmod \,2

Steenrod algebra. The description includes a procedure to obtain all of the generators, a generating function to enumerate them, and a proof of a nice conjecture about how many there are in each dimension.
A result on multidimensional quantization
Authors: Hall Eric B. , Wise Gary L.
Published: June 2010

For any integer

N
>
1

N > 1

, a probability space, a Gaussian random vector

X
X

defined on the space with a positive definite covariance matrix, and an

N
N

-level quantizer

Q
Q

are presented such that the random vector

Q
(
X
)

Q(X)

takes on each of the

N
N

values in its range with equal probability and such that

X
X

and

Q
(
X
)

Q(X)

are independent.
Undecidability of parametric solutions of polynomial equations
Authors: Kim K. H. , Roush F. W.
Published: June 2010

We prove that, for any field

F

{\mathbf {F}}

of characteristic

0
0

satisfying a hypothesis related to not being algebraically closed, the problem of finding non-constant parametric solutions in

F

(
t
)

{\mathbf {F}}(t)

to a polynomial system with coefficients in

F

{\mathbf {F}}

is algorithmically unsolvable.
An optimization of the Besicovitch covering
Authors: Loeb Peter A.
Published: June 2010

Given an appropriate covering by balls of a set in a metric space, we construct an optimized version of the subcovering used in the proof of Besicovitch’s theorem. The proof is nonstandard and suggests a general method for optimizing standard geometric constructions.
The Baire category and forcing large Lindelöf spaces with points 𝐺_{𝛿}
Authors: Gorelic Isaac
Published: June 2010

For

κ
\kappa

as large an aleph as we want, we construct by forcing a model in which CH holds and there is a Lindelöf zero-dimensional space of size

κ
\kappa

with points

G
δ

{G_\delta }

.
Ordered subrings of the reals in which output sets are recursively enumerable
Authors: Byerly Robert E.
Published: June 2010

In On a theory of computation and complexity over the real numbers ..., Bull. Amer. Math. Soc. 21 (1989), 1-46, Blum, Shub, and Smale investigated computability over the reals and over ordered rings in general. They showed that over the reals, output sets of machines are recursively enumerable (i.e., halting sets of machines). It is asked in the aforementioned paper which ordered rings have this property (which we abbreviate

O
=
R
.
E
.

O = R.E.

). In Ordered rings over which output sets are recursively enumerable, Proc. Amer. Math. Soc. 112 (1991), 569-575, Michaux characterized the members of a certain class of ordered rings of infinite transcendence degree over

Q

\mathbb {Q}

satisfying

O
=
R
.
E
.

O = R.E.

In this paper we characterize the subrings of

R

\mathbb {R}

of finite transcendence degree over

Q

\mathbb {Q}

satisfying

O
=
R
.
E
.

O = R.E.

as those rings recursive in the Dedekind cuts of members of a transcendence base. With Michaux’s result, this answers the question for subrings of

R

\mathbb {R}

(i.e., archimedean rings).
A generalization of Odlyzko’s conjecture: the coefficients of (1-𝑞)^{𝑗}/((1-𝑞²ⁿ)⋯(1-𝑞^{2𝑛+2𝑗})) alternate in sign
Authors: Friedman Jane , Zeilberger Doron
Published: June 2010

A positivity result is proved that generalizes a conjecture of Odlyzko, previously proved by Stanton and Zeilberger.
Real analytic Radon transforms on rank one symmetric spaces
Authors: Quinto Eric Todd
Published: June 2010

Using microlocal techniques, we prove support theorems for Radon transforms with real analytic measures on horocycles in rank one symmetric spaces. We generalize Helgason’s support theorem to this case and prove a new local support theorem.
Bordism classes of vector bundles over real projective spaces
Authors: Torrence Bruce F.
Published: June 2010

A basis is presented for the

R

P

n

{\mathbf {R}}{{\mathbf {P}}^n}

classes in

N

n

(
B
O
)

{\mathfrak {N}_n}(BO)

. This basis is used to prove that every smooth involution

(
M
,
T
)

(M,T)

on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.
Complete pure injectivity and endomorphism rings
Authors: Gómez Pardo J. L. , Dung Nguyen V. , Wisbauer R.
Published: June 2010

It is shown that if

M
M

is a finitely presented completely pure injective object in a locally finitely generated Grothendieck category

C

{\mathbf {C}}

such that

S
=

End

C

M

S = {\operatorname {End} _{\mathbf {C}}}M

is von Neumann regular, then

S
S

is semisimple. This is a generalized version of a well-known theorem of Osofsky, which includes also a result of Damiano on PCI-rings. As an application, we obtain a characterization of right hereditary rings with finitely presented injective hull.