bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on the divisibility of certain Chern numbers
Authors: Charitos Leonidas , Papastavridis Stavros
Published: June 2010

If

M
M

is a weakly almost complex manifold, then

c
r

(
M
)
∈

H

24

(
M
;
Z
)

{c_r}(M) \in {H^{24}}(M;Z)

is the

r
r

th Chern class of its normal bundle. Theorem 1. If

m
m

,

r
r

are natural numbers with

r
⩽
m

r \leqslant m

, then there exists a

2
m

2m

-fold

M
0

{M_0}

, compact, closed and weakly almost complex, so that the normal characteristic number

⟨

c
r

(

M
0

)

c

m
−
r

(

M
0

)

\left \langle {{c_r}({M_0}){c_{m - r}}({M_0})} \right .

,

[

M
0

]

⟩

\left . {[{M_0}]} \right \rangle

is a power of 2.
A converse to von Neumann’s inequality
Authors: Rovnyak James
Published: June 2010

The Pick-Nevanlinna theorem is used to show that if

f
0

{f_0}

is holomorphic on an open subset

G
G

of the unit disk

D
D

and

‖

f
0

(
T
)

‖

⩽
1

\left \| {{f_0}(T)} \right \| \leqslant 1

for every contraction operator

T
T

on a Hilbert space whose spectrum is contained in

G
G

, then

f
0

=
f

|

G

{f_0} = f|G

where

f
f

is holomorphic and bounded by 1 on

D
D

.
On commuting families of nonexpansive operators
Authors: Schechtman Gideon
Published: June 2010

For each

n
n

,

1
⩽
n
⩽
∞

1 \leqslant n \leqslant \infty

, there exist a weakly compact, convex subset

W
W

of

L
1

{L_1}

and a family
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace upper T Subscript i Baseline right-brace Subscript i equals 1 Superscript n Baseline left-parenthesis left-brace upper T Subscript i Baseline right-brace Subscript i equals 1 Superscript normal infinity Baseline in case n equals normal infinity right-parenthesis">

{

T
i

}

i
=
1

n

(
{

T
i

}

i
=
1

∞

 in case 

n
=
∞
)

\{ {T_i}\} _{i = 1}^n(\{ {T_i}\} _{i = 1}^\infty {\text { in case }}n = \infty )

of nonexpansive operators of

W
W

into

W
W

such that there is no common fixed point for the whole family but any (finite) strict subfamily admits a common fixed point.
Heegaard diagrams of lens spaces
Authors: Osborne R. P.
Published: June 2010

Let

(
M
,
F
;
υ
,
w
)

(M,F;\upsilon ,w)

be a Heegaard diagram of

M
M

. The complexity of this diagram is the number of points in

υ
∩
m

\upsilon \cap m

. This is also the length of the relators in a group presentation naturally corresponding to this diagram. We give an example to show that a Heegaard diagram of minimal complexity need not have a cancelling pair of meridian disks. In terms of the presentation, this says that a minimal length presentation need not have a defining relator for one of the generators. This provides a counterexample to a conjecture of Waldhausen. Our example depends on the rather trivial observation that the shortest possible

2
2

-generator presentation of the cyclic group of order

173
173

is

⟨

a
,
b

|

a

13

b
2

,

a

−
2

b

13

⟩

\left \langle {a,b|{a^{13}}{b^2},{a^{ - 2}}{b^{13}}} \right \rangle

.
Some remarks on the Hasse norm theorem
Authors: Opolka Hans
Published: June 2010

A substitute for the Hasse norm theorem in Galois extensions of number fields is made more explicit.
Plane Frobenius sandwiches
Authors: Ganong R.
Published: June 2010

Suppose the total Frobenius of an affine plane factors through a plane. Then the factorization is just the composite of relative Frobeniuses with respect to suitable coordinate axes.
Splitting universal bundles over flag manifolds
Authors: Stong R. E.
Published: June 2010

Let

F

{\mathbf {F}}

be one of the fields

R

{\mathbf {R}}

,

C

{\mathbf {C}}

, or

H

{\mathbf {H}}

and correspondingly let

F

G

{\mathbf {F}}G

be

O
O

,

U
U

, or

Sp

{\text {Sp}}

, i.e. the orthogonal, unitary, or symplectic group. Over the flag manifold
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F upper G left-parenthesis n 1 plus midline-horizontal-ellipsis plus n Subscript k Baseline right-parenthesis slash bold upper F upper G left-parenthesis n 1 right-parenthesis times midline-horizontal-ellipsis times bold upper F upper G left-parenthesis n Subscript k Baseline right-parenthesis">

F

G
(

n
1

+
⋯
+

n
k

)

/

F

G
(

n
1

)
×
⋯
×

F

G
(

n
k

)

{\mathbf {F}}G({n_1} + \cdots + {n_k})/{\mathbf {F}}G({n_1}) \times \cdots \times {\mathbf {F}}G({n_k})

one has vector bundles

γ
i

{\gamma _i}

over

F
F

of dimension

n
i

{n_i}

,

1
⩽
i
⩽
k

1 \leqslant i \leqslant k

. This paper determines all cases in which

γ
i

{\gamma _i}

decomposes nontrivially as a Whitney sum.
A system of quadrics describing the orbit of the highest weight vector
Authors: Lichtenstein Woody
Published: June 2010

Let

G
G

be a complex semisimple Lie group acting irreducibly on a finite dimensional vector space

V
V

. A simple method is given for constructing a system of quadratic equations which defines the orbit of the highest weight vector in the projective space

P
V

PV

.
Quotients of Banach spaces of cotype 𝑞
Authors: Pisier Gilles
Published: June 2010

Let

Z
Z

be a Banach space and let

X
⊂
Z

X \subset Z

be a

B
B

-convex subspace (equivalently, assume that

X
X

does not contain

l
1
n

l_1^n

’s uniformly). Then every Bernoulli series

Σ

n
=
1

∞

ε
n

z
n

\Sigma _{n = 1}^\infty {\varepsilon _n}{z_n}

which converges almost surely in the quotient

Z

/

X

Z/X

can be lifted to a Bernoulli series a.s. convergent in

Z
Z

. As a corollary, if

Z
Z

is of cotype

q
q

, then

Z

/

X

Z/X

is also of cotype

q
q

. This extends a result of [4] concerning the particular case

Z
=

L
1

Z = {L_1}

.
Bernstein’s theorem for the polydisc
Authors: Tung S. H.
Published: June 2010

A well-known theorem of Bernstein states that if a polynomial of degree

N
N

of a complex variable has its modulus no larger than one on the unit disc then the modulus of its derivative will not exceed

N
N

on the unit disc. The result has been extended to the case of polynomials on the unit ball in several complex variables. Here we generalize the theorem to the cases of the unit polydisc and the unit polycylinder which is a topological product of a unit ball and a unit polydisc.
Homological embedding properties of the fibers of a map and the dimension of its image
Authors: Walsh John J.
Published: June 2010

A relationship is established between the homological codimension of the point inverses of a map and the dimension of its image. An infinite-dimensional version leads to the conclusion that the image of a proper map defined on Hilbert space cannot be countable dimensional. A finite-dimensional version yields: if

g
:

M
n

→
Y

g:{M^n} \to Y

is a proper map,

M
n

{M^n}

is a

G
G

-orientable

n
n

-manifold without boundary, and

dim
⁡
Y
⩽
k

\dim Y \leqslant k

, then there is a point

y
∈
Y

y \in Y

and an integer

i
⩾
n
−
k

i \geqslant n - k

such that

H
ˇ

i

(

g

−
1

(
y
)
;
G
)
≠
0

\check {H}^i (g^{-1}(y);G) \ne 0

.
An infinitely divisible distribution involving modified Bessel functions
Authors: Ismail Mourad E. H. , Miller Kenneth S.
Published: June 2010

We prove that the function
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartFraction b Over a EndFraction right-parenthesis Superscript mu minus v Baseline StartFraction upper K Subscript mu Baseline left-parenthesis b x Superscript 1 slash 2 Baseline right-parenthesis upper K Subscript v Baseline left-parenthesis a x Superscript 1 slash 2 Baseline right-parenthesis Over upper K Subscript mu Baseline left-parenthesis a x Superscript 1 slash 2 Baseline right-parenthesis upper K Subscript v Baseline left-parenthesis b x Superscript 1 slash 2 Baseline right-parenthesis EndFraction">

(

b
a

)

μ
−
v

K
μ

(
b

x

1

/

2

)

K
v

(
a

x

1

/

2

)

K
μ

(
a

x

1

/

2

)

K
v

(
b

x

1

/

2

)

{\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{K_\mu }(b{x^{1/2}}){K_v}(a{x^{1/2}})}} {{{K_\mu }(a{x^{1/2}}){K_v}(b{x^{1/2}})}}

\]
is the Laplace transform of an infinitely divisible probability distribution when

v
>
μ
⩾
0

v > \mu \geqslant 0

and

b
>
a
>
0

b > a > 0

. This implies the complete monotonic ity of the function. We also establish a representation as a Stieltjes transform, which implies in particular that the function has positive real part when

x
x

lies in the right half-plane. We conjecture that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartFraction b Over a EndFraction right-parenthesis Superscript mu minus v Baseline StartFraction upper I Subscript mu Baseline left-parenthesis a x Superscript 1 slash 2 Baseline right-parenthesis upper I Subscript v Baseline left-parenthesis b x Superscript 1 slash 2 Baseline right-parenthesis Over upper I Subscript mu Baseline left-parenthesis b x Superscript 1 slash 2 Baseline right-parenthesis upper I Subscript v Baseline left-parenthesis a x Superscript 1 slash 2 Baseline right-parenthesis EndFraction">

(

b
a

)

μ
−
v

I
μ

(
a

x

1

/

2

)

I
v

(
b

x

1

/

2

)

I
μ

(
b

x

1

/

2

)

I
v

(
a

x

1

/

2

)

{\left ( {\frac {b} {a}} \right )^{\mu - v}}\frac {{{I_\mu }(a{x^{1/2}}){I_v}(b{x^{1/2}})}} {{{I_\mu }(b{x^{1/2}}){I_v}(a{x^{1/2}})}}

\]
also is the Laplace transform of an infinitely divisible probability distribution. It is known that in the limit as

v
→
∞

v \to \infty

, the infinite divisibility property holds for both functions.
A Krasnosel′skiĭ-type theorem for points of local nonconvexity
Authors: Breen Marilyn
Published: June 2010

Let

S
S

be a compact connected set in

R
2

{R^2}

,

S
S

not convex. Then

S
S

is starshaped if and only if every 3 points of local nonconvexity of

S
S

are clearly visible from a common point of

S
S

. For

k
=
1

k = 1

or

k
=
2

k = 2

, dimker

S
⩾

S \geqslant

k
k

if and only if for some

∈>
0

\in > 0

, every

f
(
k
)
=
max

{

3
,
6
−
2
k

}

f(k) = \max \left \{ {3,6 - 2k} \right \}

points of local nonconvexity of

S
S

are clearly visible from a common

k
k

-dimensional

∈
\in

neighborhood in

S
S

. Each result is best possible.
Topological entropy of foliation preserving diffeomorphisms
Authors: Yano Koichi
Published: June 2010

We give some estimations for the topolotical entropy of diffeomorphisms preserving foliated structures, one of which partially supports the entropy conjecture.
Intersections of Γ-isotype subgroups in abelian groups
Authors: Bečvář Jindřich
Published: June 2010

A subgroup

H
H

of an abelian group

G
G

is an intersection of isotype subgroups of

G
G

if and only if, for each prime

p
p

, if

x
+
H

x + H

is a coset of order

p
p

then there is another coset of order

p
p

containing an element

y
y

of order

p
p

such that

h
p
∗

(
x
)
⩽

h
p
∗

(
y
)

h_p^* (x) \leqslant h_p^*(y)

. A subgroup

H
H

of

G
G

is isotype in

G
G

if and only if, for each prime

p
p

, every coset

x
+
H

x + H

of order

p
p

contains an element y of order

p
p

such that

h
p
∗

(
x
)

h
p
∗

(
y
)

h_p^*(x)h_p^*(y)
Cauchy conditions on symmetrics
Authors: Davis S. W.
Published: June 2010

We call a symmetric

d
d

on a space

X
X

a

wC

{\text {wC}}

symmetric if whenever

A
⊆
X

A \subseteq X

and there exists

ε
>
0

\varepsilon > 0

such that

d
(
x
,
y
)
⩾
ε

d(x,y) \geqslant \varepsilon

for all

x
x

,

y
∈
A

y \in A

, then

A
A

is relatively discrete. We show that there are no

L
L

-spaces which admit

wC

{\text {wC}}

symmetries. The

wC

{\text {wC}}

notion is extended to certain weaker structures such as

F

\mathcal {F}

-spaces with similar results.
Dieudonné-Schwartz theorem on bounded sets in inductive limits. II
Authors: Kučera J. , Bosch C.
Published: June 2010

The Dieudonné-Schwartz Theorem [1, Chapter 2, §12] has been stated for strict inductive limits. In [3] it has been extended to inductive limits. Here the result of [3] is generalized. Also, the case when each set bounded in ind lim

E
n

{E_n}

is contained, but not necessarily bounded, in some

E
n

{E_n}

is considered.
Semiregular invariant measures on abelian groups
Authors: Pelc Andrzej
Published: June 2010

A nonnegative countably additive, extended real-valued measure is called semiregular if every set of positive measure contains a set of positive finite measure. V. Kannan and S. R. Raju [3] stated the problem of whether every invariant semiregular measure defined on all subsets of a group is necessarily a multiple of the counting measure. We prove that the negative answer is equivalent to the existence of a real-valued measurable cardinal. It is shown, moreover, that a counterexample can be found on every abelian group of real-valued measurable cardinality.
Unit lemniscates contained in the unit ball
Authors: Shih Mau Hsiang , Wang Hann Tzong
Published: June 2010

Let

{

A
1

,

A
2

,
…
,

A
ν

}
≡
A

\{ {A_1},{A_2}, \ldots ,{A_\nu }\} \equiv A

be a set of points in

E
n

{E^n}

. Let

E
(
A
)

E(A)

be the set of points in

E
n

{E^n}

such that

Π

k
=
1

ν

p

A
k

¯

⩽
1

\Pi _{k = 1}^\nu \overline {p{A_k}} \leqslant 1

(where

p

A
k

¯

\overline {p{A_k}}

denotes the Euclidean distance between

p
p

and

A
k

{A_k}

), and call this set the unit lemniscate with focal set

A
A

. It is shown that if the vertices of a regular tetrahedron lie at the distance

δ
∈
(
0
,
1
)

\delta \in (0,1)

from the origin, then they are the foci of a unit lemniscate contained in the open unit ball of

E
3

{E^3}

if and only if the sign of

36
−
64
δ
+
2

δ
2

−
64

δ
3

+
36

δ
4

+
27

δ
6

36 - 64\delta + 2{\delta ^2} - 64{\delta ^3} + 36{\delta ^4} + 27{\delta ^6}

is positive.
On quasinilpotent semigroups of operators
Authors: Guinand P. S.
Published: June 2010

We construct a pair of operators such that the semigroup generated by them consists of operators which are nilpotent of index 3. The sum of the two operators, however, is not quasinilpotent.
Simplified 𝐿^{∞} estimates for difference schemes for partial differential equations
Authors: Layton William J.
Published: June 2010

L
∞

{L^\infty }

estimates for one-step difference approximations to the Cauchy problem for

∂
u

/

∂
t
=
∂
u

/

∂
x

\partial u/\partial t = \partial u/\partial x

are proven by means of simple

L
2

{L^2}

-techniques. It is shown that, provided the difference approximation is stable in

L
2

{L^2}

(and not necessarily

L
∞

{L^\infty }

) and accurate of order

r
r

, the error in approximating smooth solutions is

O
(

h
r

)

O({h^r})

. This has been proven by Hedstrom and Thomée using Fourier multipliers and Besov spaces. The present paper shows how convergence rates in

L
∞

{L^\infty }

can be recovered using simple techniques (such as the Fourier inversion formula). The methods of Hedstrom and Thomée give sharper results when the difference scheme diverges. The present paper exploits the fact that estimates between

L
∞

{L^\infty }

and

L
1

{L^1}

are frequently easier to obtain than between

L
∞

{L^\infty }

and

L
∞

{L^\infty }

.
Totally real minimal submanifolds in a complex projective space
Authors: Ejiri Norio
Published: June 2010

We give a pinching theorem with respect to the scalar curvatures of

4
4

-dimensional conformally flat totally real minimal submanifolds in a

4
4

-dimensional complex projective space.
More on M. E. Rudin’s Dowker space
Authors: Hart Klaas Pieter
Published: June 2010

It is shown that M. E. Rudin’s Dowker space is finitely-fully normal and orthocompact, thus answering questions of Mansfield and Scott.
Finite 2-complexes with infinitely-generated groups of self-homotopy-equivalences
Authors: Brunner A. M. , Ratcliffe J. G.
Published: June 2010

Examples of finite

2
2

-dimensional aspherical cell complexes are given whose group of homotopy classes of self-homotopy-equivalences is infinitely generated.
A theory of interval iteration
Authors: Rall L. B.
Published: June 2010

A theory of interval iteration, based on a few simple assumptions, is given for the fixed point problem for operators in partially ordered topological spaces. A comparison of interval with ordinary iteration is made which shows that their properties are converse in a certain sense with respect to existence or nonexistence of fixed points. The theory of interval iteration is shown to hold without modification if the computation is restricted to a finite set of points, as in actual practice. In this latter case, interval iteration is shown to converge or diverge in a finite number of steps, for which an upper bound is given. By the introduction of a suitable iteration operator, the method of interval iteration is extended to the problem of solution of equations in linear spaces.
Taming compacta in 𝐸⁴
Authors: Walsh John J. , Wright David G.
Published: June 2010

A compactum

X
X

in Euclidean

4
4

-space

E
4

{E^4}

is shown to be tame if its projection into

E
3

{E^3}

is

1
1

-dimensional and if

dim
⁡
L
∩
X
⩽
0

\dim L \cap X \leqslant 0

for each vertical line

L
L

in

E
4

{E^4}

. An essential ingredient is the result due to J. L. Bryant and D. L. Sumners that a

1
1

-dimensional compactum in a

3
3

-dimensional hyperplane of

E
4

{E^4}

is tame in

E
4

{E^4}

.
Locally contractive and expansive mappings
Authors: Hu Thakyin , Rosen Harvey
Published: June 2010

Locally contractive and locally expansive mappings are studied in this paper. A remetrization theorem and some fixed point theorems are proved.
On the extension of 𝐻^{𝑝}-functions in polydiscs
Authors: Chee P. S.
Published: June 2010

For

N
=
2

or

3

N = 2\;{\text {or}}\;3

it is shown that if

E
E

is the zero set of a holomorphic function in

U
N

{U^N}

satisfying the separation condition of Alexander [1], viz., there exist

r
∈
(
0
,
1
)

r \in (0,1)

and

δ
>
0

\delta > 0

such that

|

α
−
β

|

⩾
δ

\left | {\alpha - \beta } \right | \geqslant \delta

whenever

(

z
′

,
α
,
z
)
≠
(

z
′

,
β
,
z
)

(z’,\alpha ,z) \ne (z’,\beta ,z)

are both in

(

Q

k
−
1

×
U
×

Q

N
−
k

)
∩
E

({Q^{k - 1}} \times U \times {Q^{N - k}}) \cap E

, where

Q
=
{
λ
∈

C

:
r
>

|
λ
|

>
1
}

Q = \{ \lambda \in {\mathbf {C}}:r > \left | \lambda \right | > 1\}

, then (a)

E
E

is the zero set of some

F
∈

H
∞

(

U
N

)

F \in {H^\infty }({U^N})

, and (b)

0
>
p
⩽
∞

0 > p \leqslant \infty

, every

g
∈
H
(
E
)

g \in H(E)

such that

|
g
|

p

{\left | g \right |^p}

has a pluriharmonic majorant on

E
E

extends to a

G
∈

H
p

(

U
N

)

G \in {H^p}({U^N})

. This generalizes earlier results of the author [3] and Zarantonello [9].
On the degree of approximation of a class of functions by means of Fourier series
Authors: Mazhar S. M.
Published: June 2010

In this paper degree of approximation of Lebesgue integrable functions by means of Fourier series is examined.
Upper semicontinuous collections of continua in class 𝑊
Authors: Proctor C. Wayne
Published: June 2010

A continuum is proven to be in Class

W
W

if it can be decomposed into an upper semicontinuous collection of

C
C

-sets, each of which is contained in Class

W
W

, and if the upper semicontinuous decomposition space thus formed is in Class

W
W

.
The ruled residue theorem for simple transcendental extensions of valued fields
Authors: Ohm Jack
Published: June 2010

A proof is given for the Ruled Residue Conjecture, which asserts that if

υ
\upsilon

is a valuation of a simple transcendental field extension

K
0

(
x
)

{K_0}(x)

and

υ
0

{\upsilon _0}

is the restriction of

υ
\upsilon

to

K
0

{K_0}

, then the residue field of

υ
\upsilon

is either ruled or algebraic over the residue field of

υ
0

{\upsilon _0}

.
Minimal surfaces with constant curvature in 4-dimensional space forms
Authors: Kenmotsu Katsuei
Published: June 2010

We classify minimal surfaces with constant Gaussian curvature in a

4
4

-dimensional space form without any global assumption. As a corollary of the main theorem, we show there is no isometric minimal immersion of a surface with constant negative Gaussian curvature into the unit

4
4

-sphere even locally. This gives a partial answer to a problem proposed by S. T. Yau.
A counterexample to the generalized Banach theorem
Authors: Bzyl Włodzimierz
Published: June 2010

We show that it is consistent that the family of Borel maps of class 2 differs from the family of pointwise limits of Borel maps of class 1. This gives an answer to a question raised by W. G. Fleissner.
Convolution operators of weak type (𝑝,𝑝) which are not of strong type (𝑝,𝑝)
Authors: Szwarc Ryszard
Published: June 2010

We give an example of a locally compact group

G
G

for which, for every

p
p

with

2
>
p
>
∞

2 > p > \infty

, there exists an operator of weak type

(
p
,
p
)

(p,p)

commuting with the right translations on

G
G

which is not of strong type

(
p
,
p
)

(p,p)

. This gives a negative solution of E. M. Stein’s problem.
One example of the boundary behaviour of biholomorphic transformations
Authors: Fridman B. L.
Published: June 2010

Two biholomorphically equivalent domains

Ω
1

{\Omega _1}

,

Ω
2

⊂

C

2

{\Omega _2} \subset {{\mathbf {C}}^2}

with piecewise smooth boundaries and with the following property are constructed. If

F
:

Ω
1

→

Ω
2

F:{\Omega _1} \to {\Omega _2}

is any biholomorphic transformation then neither

F
F

nor

F

−
1

{F^{ - 1}}

can be extended continuously to the boundary.
Some applications of generalized exponentials to partial differential equations
Authors: Abramowich J.
Published: June 2010

Using what may be considered as a natural generalization of the exponential function, some of the formalism of the theory of ordinary linear differential equations is extended to a class of linear partial differential equations among which are some important equations of mathematical physics. In

\S 

1

{\text {\S }}1

we give the definitions of the generalized exponentials and derive expressions for them.

§
2

\S 2

is devoted to the study of some of the properties of the exponential in two independent variables. In

§
3

\S 3

we derive the general solutions of some key partial differential equations using the method of recursion. The last section is devoted to extending the formalism of the method of variation of parameters to a class of linear partial differential equations.
A conjecture on compact Fréchet spaces
Authors: Zhou Hao Xuan
Published: June 2010

Let

X
X

be a compact Hausdorff space.

X
X

is Fréchet if every feebly compact subset is closed in

X
X

. Under MA, the converse is false.
Monotonic semiconjugacies onto expanding maps of the interval
Authors: Byers Bill
Published: June 2010

A contraction mapping is used to produce a semiconjugacy from a map

τ
1

{\tau _1}

with maximum at

c
1

{c_1}

to an expanding unimodal map with maximum at

c
2

{c_2}

under the assumption that there is an interval

J
J

containing

c
1

{c_1}

such that there is a one-to-one, order-preserving correspondence between the orbit of

J
J

under

τ
1

{\tau _1}

and the orbit of

c
2

{c_2}

under the expanding map.
Monomial equimultiple curves in positive characteristic
Authors: Narasimhan R.
Published: June 2010

It is known that the local equimultiple locus of a hypersurface in characteristic zero is contained in a regular hypersurface. Here we give an example of a monomial curve on a threefold in positive characteristic

p

 > 

2

p{\text { > }}2

which is equimultiple but not hyperplanar. As a corollary we have that any monomial curve which lies on a certain type of hypersurface (whose local equation is of a special form in its natural

p
p

-basis expression) is automatically equimultiple for the hypersurface.
The Freiheitssatz for one-relation monoids
Authors: Squier C. , Wrathall C.
Published: June 2010

We give an elementary proof of the Freiheitssatz for one-relation monoids.
Extension of uniform measures
Authors: Caby Errol
Published: June 2010

Necessary and sufficient conditions for a uniform measure to have a tight extension are presented. From these results, conditions for the factors of a tight measure to be also tight are derived.
Persistently finite, persistently arithmetic theories
Authors: Ash C. J. , Millar T. S.
Published: June 2010

The main result shows that a certain class of theories, the persistently finite, persistently arithmetic theories, have only arithmetic countable models.
Countable products of scattered paracompact spaces
Authors: Rudin Mary Ellen , Watson Steve
Published: June 2010

In this paper we prove that the product of countably many scattered paracompact spaces is even ultraparacompact.
On the strong unicity of best Chebyshev approximation of differentiable functions
Authors: Kroó András
Published: June 2010

Let

X
X

be a normed linear space,

U
n

{U_n}

an

n
n

-dimensional Chebyshev subspace of

X
X

. For

f
∈
X

f \in X

denote by

p
(
f
)
∈

U
n

p(f) \in {U_n}

its best approximation in

U
n

{U_n}

. The problem of strong unicity consists in estimating how fast the nearly best approximants

g
∈

U
n

g \in {U_n}

satisfying

‖

f
−
g

‖

⩽

‖

f
−
p
(
f
)

‖

+
δ

\left \| {f - g} \right \| \leqslant \left \| {f - p(f)} \right \| + \delta

approach

p
(
f
)

p(f)

as

δ
→
0

\delta \to 0

. In the present note we study this problem in the case when

X
=

C
r

[
a
,
b
]

X = {C^r}[a,b]

is the space of

r
r

-times continuously differentiable functions endowed with the supremum norm

(
1
⩽
r
⩽
∞
)

(1 \leqslant r \leqslant \infty )

.
A note on the rotation number of Poincaré
Authors: Yu Shu Xiang
Published: June 2010

The present note gives a formula relating the rotation number, the conjugating function and the vector field for a flow on a torus. Furthermore, in a particular case, it gives a formula such that the rotation number

ρ
\rho

can be computed only by means of the vector field

f
(
x
,
y
)

f(x,y)

.
The Lévy-Lindeberg central limit theorem in Orlicz spaces 𝐿_{Φ}
Authors: Ławniczak Anna T.
Published: June 2010

An

L
ϕ

(
T
,

F

,
m
)

{L_\phi }(T,\mathcal {F},m)

-valued random element

X
X

, where

Φ
(

t

1

/

2

)

\Phi ({t^{1/2}})

is equivalent to a concave function, satisfies the Lévy-Lindeberg central limit theorem if and only if it is centered and pre-Gaussian; that is, if and only if

E
X
(
t
)
=
0

EX(t) = 0

m
m

-a.e. and

{
E

X
2

(
t
)

}

1

/

2

∈

L
ϕ

{\{ E{X^2}(t)\} ^{1/2}} \in {L_\phi }

.
Lie ideals and Jordan derivations of prime rings
Authors: Awtar Ram
Published: June 2010

Herstein proved [1, Theorem 3.3] that any Jordan derivation of a prime ring of characteristic not 2 is a derivation of

R
R

. Our purpose is to extend this result on Lie ideals. We prove the following Theorem. Let

R
R

be any prime ring such that char

R
≠
2

R \ne 2

ana let

U
U

be a Lie ideal of

R
R

such that

u
2

∈
U

{u^2} \in U

for all

u
∈
U

u \in U

. If ,’, is an additive mapping of

R
R

into itself satisfying

(

u
2

)
′

=

u
′

u
+
u

u
′

({u^2})’ = u’u + uu’

for all

u
∈
U

u \in U

, then

(
u
υ

)
′

=

u
′

υ
+
u

υ
′

(u\upsilon )’ = u’\upsilon + u\upsilon ’

for all

u
,
υ
∈
U

u,\upsilon \in U

.
Co-well-powered reflective subcategories
Authors: Hoffmann Rudolf-E.
Published: June 2010

A full isomorphism-closed subcategory

A

\mathcal {A}

of a complete well-powered and co-well-powered category

C

\mathcal {C}

is both co-well-powered (in its own right) and reflective in

C

\mathcal {C}

if and only if (a)

A

\mathcal {A}

is closed in

C

\mathcal {C}

under the formation of (

U
U

-small-indexed) limits, and (b) the epi-reflective hull

B

\mathcal {B}

of

A

\mathcal {A}

in

C

\mathcal {C}

is co-well-powered.
Tales about tails
Authors: Waadeland Haakon
Published: June 2010

Necessary and sufficient conditions are given for a sequence

{

g

(
n
)

}

\left \{ {{g^{(n)}}} \right \}

to be a sequence of tails of a convergent continued fraction. Some special cases are also studied.
Oscillation theorems for 𝑛th order nonlinear differential equations with deviating arguments
Authors: Grace S. R. , Lalli B. S.
Published: June 2010

In this note we study the oscillatory behavior of solutions of the

n
n

th order nonlinear functional differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Superscript left-parenthesis n right-parenthesis Baseline left-parenthesis t right-parenthesis plus q left-parenthesis t right-parenthesis f left-parenthesis x left-bracket g left-parenthesis t right-parenthesis right-bracket right-parenthesis equals 0 comma n even comma">

x

(
n
)

(
t
)
+
q
(
t
)
f
(
x

[

g
(
t
)

]

)
=
0
,

n

even,

{x^{(n)}}(t) + q(t)f(x\left [ {g(t)} \right ]) = 0,\quad n\;{\text {even,}}

\]
without assuming that the deviating argument is retarded or advanced. Sufficient conditions are established for all solutions of the equation to be oscillatory.
Tarski’s extension theorem for group-valued charges
Authors: Aversa Vincenzo , Rao K. P. S. Bhaskara
Published: June 2010

A result of Tarski on extensions of real-valued charges is extended to group-valued charges for certain groups.
Almost-quaternion (𝑚-1)-substructures on 𝑆^{4𝑚-3}
Authors: Önder Turgut
Published: June 2010

We prove that

S

4
m
−
3

{S^{4m - 3}}

admits an almost-quaternion

(
m
−
1
)

(m - 1)

-substructure if and only if

m
=
2

m = 2

, completing the missing case in our paper On quaternionic James numbers and almost-quaternion substructures on the sphere.
Spectral inclusion for subnormal 𝑛-tuples
Authors: Putinar Mihai
Published: June 2010

Let

S
S

be a subnormal operator on a Hilbert space and let

N
N

be its minimal extension. Then a celebrated theorem due to P. Halmos asserts that

Sp

(
N
)
⊂

Sp

(
S
)

{\text {Sp}}(N) \subset {\text {Sp}}(S)

, denoting by

S

P

{{\text {S}}_{\text {P}}}

the spectrum. This note contains a multidimensional version, with respect to Taylor’s joint spectrum, of this spectral inclusion theorem.
A stable group-pair invariant of three-dimensional manifolds
Authors: Rolfsen Dale
Published: June 2010

A new algebraic invariant of

3
3

-manifolds is discussed. It is an application of the rational surgery calculus.
Atriodic acyclic continua and class 𝑊
Authors: Davis James F.
Published: June 2010

A continuum

M
M

is in class

W
W

provided that for each continuum

Y
Y

and mapping

f
f

of

Y
Y

onto

M
M

, each subcontinuum of

M
M

is the image under

f
f

of some subcontinuum of

Y
Y

. It is shown that atriodic continua with trivial first Čech cohomology are in class

W
W

.
Positive definite matrices and Catalan numbers, revisited
Authors: Shapiro Louis W.
Published: June 2010

In this note a combinatorial correspondence is used to prove that the number of positive definite, tridiagonal, integral matrices of determinant 1 whose sub and super diagonals consist solely of ones is

C
n

=

(
n

2
n

)

/

(
n
+
1
)

{C_n} = (_n^{2n})/(n + 1)

. The correspondence is then further used to count such matrices by trace and also by number of ones on the main diagonal. Other related correspondences and results are given including those for determinant equal to

2
,
3
,
4

a
n
d
5

2,3,4{\rm {and5}}

.
A note on disjointness preserving operators
Authors: de Pagter B.
Published: June 2010

In this paper we present some results concerning the automatic order boundedness of disjointness preserving operators on Riesz spaces (vector lattices).
A characterization of spectral operators on Hilbert spaces
Authors: Tanahashi Kôtarô , Yoshino Takashi
Published: June 2010

In [8] Wadhwa shows that if a bounded linear operator

T
T

on a complex Hilbert space

H
H

is a decomposable operator and has the condition (I), then

T
T

is a spectral operator with a normal scalar part. In this paper, by using this result, we show that a weak decomposable operator

T
T

is a spectral operator with a normal scalar part if and only if

T
T

satisfies the assertion that (1)

T
T

has the conditions (

C
C

) and (

I
I

) or that (2) every spectral maximal space of

T
T

reduces

T
T

. This result improves [1, 6 and 7]. From this result, we can get a characterization of spectral operators, but this result does not hold in complex Banach space (see Remark 2).
The ultrafilter characterization of huge cardinals
Authors: Mignone Robert J.
Published: June 2010

A huge cardinal can be characterized using ultrafilters. After an argument is made for a particular ultrafilter characterization, it is used to prove the existence of a measurable cardinal above the huge cardinal, and an ultrafilter over the set of all subsets of this measurable cardinal of size smaller than the huge cardinal. Finally, this last ultrafilter is disassembled intact by a process which often produces a different ultrafilter from the one started out with. An important point of this paper is given the existence of the particular ultrafilter characterization of a huge cardinal mentioned above these results are proved in Zermelo-Fraenkel set theory without the axiom of choice.
Each 𝑅^{∞}-manifold has a unique piecewise linear 𝑅^{∞}-structure
Authors: Sakai Katsuro
Published: June 2010

R. E. Heisey introduced piecewise linear

R

∞

{{\mathbf {R}}^\infty }

-structures and defined piecewise linear

R

∞

{{\mathbf {R}}^\infty }

-manifolds. In this paper we show that two piecewise linear

R

∞

{{\mathbf {R}}^\infty }

-manifolds are isomorphic if they have the same homotopy type. From the Open Embedding Theorem for (topological)

R

∞

{{\mathbf {R}}^\infty }

-manifolds and this result, we have the title.
A remark on refinable maps and calmness
Authors: Kato Hisao
Published: June 2010

It is shown that if

r
:
X
→
Y

r:X \to Y

is a refinable map between compacta and

Y
Y

is calm, then

r
r

is a shape equivalence. As a corollary, if

r
:
X
→
Y

r:X \to Y

is a refinable map between compacta and either

X
X

or

Y
Y

is

S
n

{S^n}

-like

(
n
⩾
1
)

(n \geqslant 1)

, then

r
r

is a shape equivalence, where

S
n

{S^n}

denotes the

n
n

-sphere.
Nonclassical Godeaux surfaces in characteristic five
Authors: Miranda Rick
Published: June 2010

A classical Godeaux surface is a smooth minimal projective surface

X
X

, with

K
X
2

=
1

K_X^2 = 1

,

p
a

=

p
g

=
0

{p_a} = {p_g} = 0

and

Pi

c

τ

(
X
)
=

Z

/

5

Z

{\text {Pi}}{{\text {c}}^\tau }(X) = {\mathbf {Z}}/5{\mathbf {Z}}

. A nonclassical Godeaux surface is a smooth minimal projective surface

X
X

with

K
X
2

=
1

K_X^2 = 1

,

p
a

=
0

{p_a} = 0

,

p
g

=
1

{p_g} = 1

and

Pi

c

τ

(
X
)
=

μ
5

{\text {Pi}}{{\text {c}}^\tau }(X) = {\mu _5}

or

α
5

{\alpha _5}

; such surfaces should exist in characteristic 5. It is the purpose of this note to construct nonclassical Godeaux surfaces in characteristic 5, with

Pi

c

τ

(
X
)
=

μ
5

{\text {Pi}}{{\text {c}}^\tau }(X) = {\mu _5}

. The method is to exhibit a smooth quintic surface on which

Z

/

5

Z

{\mathbf {Z}}/5{\mathbf {Z}}

acts, so that the quotient is smooth; this quotient is the desired surface.
Equivalence of the Green’s functions for diffusion operators in 𝑅ⁿ: a counterexample
Authors: Bauman Patricia
Published: June 2010

In a smooth domain in

R

n

{{\mathbf {R}}^n}

, the Green’s functions for second-order, uniformly elliptic operators in divergence form are all proportional to the Green’s function for the Laplacian [7]. In this paper we show that the above result fails for diffusion operators, that is, second-order, uniformly elliptic operators with continuous coefficients in nondivergence form. In fact, we give an example in which the Green’s function is locally unbounded away from the pole.
Criteria for closedness of vector measures
Authors: Ricker W.
Published: June 2010

It is shown that there is a large class of locally convex spaces, which includes, for example, all spaces which are metrizable or Suslin or have the strict Mackey convergence property as well as many dual spaces, with the property that any vector measure assuming its values in such a space is a closed measure.
A generalization of the Sierpiński theorem
Authors: Dijkstra J. J.
Published: June 2010

Sierpiński’s theorem admits the following generalization. Let

n
n

be a nonnegative integer and

X
X

a compact Hausdorff space. If

{

F
i

|

i
∈

N

}

\left \{ {{F_i}\left | {i \in {\mathbf {N}}} \right .} \right \}

is a countable closed covering of

X
X

such that

(

F
i

∩

F
j

)

>
n

\left ( {{F_i} \cap {F_j}} \right ) > n

for distinct

i
i

and

j
j

in

N

{\mathbf {N}}

, then every continuous mapping from

F
1

{F_1}

into the

n
n

-sphere

S
n

{S^n}

is extendable over

X
X

.
The dual ring to a coring with a grouplike
Authors: Kleiner Mark
Published: June 2010

A matrix problem is that of the classification up to isomorphism of induced comodules over a coring with a grouplike. To define such a problem, using a more familiar language of induced modules, one wants to know what ring is dual to a coring with a grouplike. The answer is given in this paper.
A comparison theorem for solutions of stochastic differential equations and its applications
Authors: Huang Zhi Yuan
Published: June 2010

A new kind of comparison theorem in which an SDE is compared with two deterministic ODEs is established by means of the generalized sample solutions of SDEs. Using this theorem, we can compare solutions of two SDEs with different diffusion coefficients and obtain some asymptotic estimations for the paths of diffusion processes.
On collectionwise normality of product spaces. I
Authors: Chiba Keiko
Published: June 2010

In this paper the following result will be obtained: Let

X
X

be a collectionwise normal

Σ
\Sigma

-space (in the sense of Nagami [9]) and

Y
Y

a paracompact first countable

P
P

-space (in the sense of Morita [8]). Then

X
×
Y

X \times Y

is collectionwise normal.
Estimates for nonanalytic cusp forms
Authors: Johnson David
Published: June 2010

Estimates for the magnitude of nonanalytic cusp forms are obtained via the theory of the Selberg transformation and Fourier coefficient estimates.
A characterization of algebras of invariant-coinvariant module type
Authors: Makino Ryohei
Published: June 2010

K. R. Fuller has characterized rings of left invariant module type. An algebra is said to be of invariant-coinvariant module type if each of its indecomposable modules is quasi-injective or quasi-projective. In this note we shall give a characterization of algebras of invariant-coinvariant module type, which distinguishes this class from that of algebras of local-colocal type. It seems of interest that the distributivity of second radicals of primitive ideals appears in our characterization.
On disconjugacy and 𝑘-point disconjugacy for linear ordinary differential operators
Authors: Muldowney James S.
Published: June 2010

It is shown that a linear ordinary differential operator of order

n
n

which is

k
k

-point disconjugate on an open interval for some

k
k

,

2
≤
k
≤
n

2 \le k \le n

, is also disconjugate on that interval.
A note on the Lusin-Privalov radial uniqueness theorem and its converse
Authors: Berman Robert D.
Published: June 2010

For

f
f

meromorphic on

Δ
\Delta

, let

f
∗

{f^ * }

denote the radial limit function of

f
f

, defined at each point of

M

R

{\mathcal {M}_R}

where the limit exists. Let

M

R

{\mathcal {M}_R}

denote the class of functions for which

f
∗

{f^ * }

exists in a residual subset of

C
C

. We prove the following theorem closely related to the Lusin-Privalov radial uniqueness theorem and its converse. There exists a nonconstant function

f
f

in

M

R

{\mathcal {M}_R}

such that

f
∗

(
η
)

=
0

{f^ * }\left ( \eta \right ) = 0

,

η
∈
E

\eta \in E

, if and only if

E
E

is not metrically dense in any open arc of

C
C

. We then show that sufficiency can be proved using functions whose moduli have radial limits at each point of

C
C

.
Functions of well-bounded operators
Authors: Benzinger Harold E.
Published: June 2010

It is shown that if

A
A

is a well-bounded operator of type (B), and if

f
f

is of bounded variation and piecewise monotone, then

f
(
A
)

f(A)

is also well bounded of type (B).
On nilpotent and solvable Malcev algebras
Authors: Stitzinger Ernest L.
Published: June 2010

Solvable and nilpotent Malcev algebras are investigated in the spirit of extending results from Lie algebras. The first section presents a proof of Engel’s theorem with the Jacobson refinement and some consequences. The second part deals with conjugacy of Cartan subalgebras and consequences.
Krull and global dimensions of fully bounded Noetherian rings
Authors: Brown Kenneth A. , Warfield R. B.
Published: June 2010

The main result of this paper states that the Krull dimension of a fully bounded Noetherian ring containing an uncountable central subfield is bounded above by its global dimension, provided that the latter is finite.
Linear maps between certain nonseparable 𝐶*-algebras
Authors: Huruya Tadasi
Published: June 2010

There exists a noninjective commutative

C
∗

{C^ * }

-algebra

A
A

such that every bounded linear map of any

C
∗

{C^ * }

-algebra into

A
A

is decomposed as a linear combination of positive linear maps.
A characterization of the Weyl spectrum
Authors: Pokrzywa Andrzej
Published: June 2010

It is shown that for each closed subset

Ω
\Omega

of the semi-Fredholm domain of a bounded linear operator

T
T

acting in a complex Hilbert space

H
H

there exists a subspace of a finite codimension in

H
H

such that the compression of

T
−
λ

T - \lambda

to this subspace is a left- or right-invertible operator for all

λ
\lambda

in

Ω
\Omega

. From this result we obtain a characterization of the Weyl spectrum of

T
T

.
A noncompletely continuous operator on 𝐿₁(𝐺) whose random Fourier transform is in 𝑐₀(𝐺̂)
Authors: Ghoussoub N. , Talagrand M.
Published: June 2010

Let

T
T

be a bounded linear operator from

L
1

(
G
,
λ
)

{L_1}(G,\lambda )

into

L
1

(
Ω
,

F

,
P
)

{L_1}(\Omega ,\mathcal {F},P)

, where

(
G
,
λ
)

(G,\lambda )

is a compact abelian metric group with its Haar measure, and

(
Ω
,

F

,
P
)

(\Omega ,\mathcal {F},P)

is a probability space. Let

(

μ
ω

)

({\mu _\omega })

be the random measure on

G
G

associated to

T
T

; that is,

T
f
(
ω
)
=

∫
G

f
(
t
)
d

μ
ω

(
t
)

Tf(\omega ) = \int _G {f(t)d{\mu _\omega }(t)}

for each

f
f

in

L
1

(
G
)

{L_1}(G)

. We show that, unlike the ideals of representable and Kalton operators, there is no subideal

B
B

of

M

(
G
)

\mathcal {M}(G)

such that

T
T

is completely continuous if and only if

μ
ω

∈
B

{\mu _\omega } \in B

for almost

ω
\omega

in

Ω
\Omega

. We actually exhibit a noncompletely continuous operator

T
T

such that

μ
^

ω

∈

l

2
+
ε

(

G
^

)

{\hat \mu _\omega } \in {l_{2 + \varepsilon }}(\hat G)

for each

ε
>
0

\varepsilon > 0

.
Singular functions and division in 𝐻^{∞}+𝐶
Authors: Gorkin Pamela B.
Published: June 2010

In this paper it is shown that for each inner function

u
u

, there exists a singular inner function

S
S

which is divisible in

H
∞

+
C

{H^\infty } + C

by all positive powers of

u
u

.
A note on hyponormal weighted shifts
Authors: Fan Peng
Published: June 2010

If

T
T

is a hyponormal weighted shift (unilateral or bilateral) and if

p
p

is a polynomial,

p
(
T
)

p(T)

may not be hyponormal.
Noetherian rings with free additive groups
Authors: O’Neill John D.
Published: June 2010

There exists a Noetherian ring of arbitrarily large cardinality with free additive group. This answers a problem of L. Fuchs.
A new proof of the Ran-Matsusaka criterion for Jacobians
Authors: Collino Alberto
Published: June 2010

We give a characteristic free proof of a criterion which was first proved by Ran over

C

{\mathbf {C}}

.
Combinatorial examples in universal coalgebra. III
Authors: Davis Robert C.
Published: June 2010

Categories cotripleable over the category of sets, for which the right adjoint at 2 has six elements and the terminal object more than one element, are surveyed along with some related categories. Some are related to multigraphlike structures, and others to sets acted upon by monoids.
Functionals of rational type over the class 𝑆
Authors: Brickman Louis
Published: June 2010

Let

L
L

be a continuous linear functional on the space of functions holomorphic in the unit disk, and let

f
f

be a function in the class

S
S

for which Re

L
L

achieves its maximum on

S
S

. Then

L
L

is said to be of rational type if the expression

L
(

f
2

/

(
f
−
w
)
)

L({f^2}/(f - w))

, which occurs in Schiffer’s differential equation, is a rational function of

w
w

. Various equivalent formulations of "rational type" are found and an application to the process of arc truncation of support points of

S
S

is made.
Regularity of Banach algebras generated by analytic semigroups satisfying some growth conditions
Authors: Esterle J. , Galé J. E.
Published: June 2010

We show that if a commutative complex Banach algebra

A
A

is generated by a nonzero analytic semigroup

(

a
t

)
Re
⁡
t
>
0

({a^t})\operatorname {Re} t > 0

satisfying
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral StartLayout 1st Row plus normal infinity 2nd Row negative normal infinity EndLayout StartFraction log Superscript plus Baseline double-vertical-bar a Superscript 1 plus i t Baseline double-vertical-bar Over 1 plus t squared EndFraction d t greater-than plus normal infinity comma">

∫

+
∞

−
∞

log

+

‖

a

1
+
i
t

‖

1
+

t
2

d
t
>
+
∞
,

\int {\begin {array}{*{20}{c}} { + \infty } \\ { - \infty } \\ \end {array} } \frac {{{{\log }^ + }\left \| {{a^{1 + it}}} \right \|}}{{1 + {t^2}}}dt > + \infty ,

\]
, then

A
A

is regular in Shilov’s sense.
An integral formula for multidimensional evolution equations
Authors: Svendsen E. C.
Published: June 2010

This paper contains an integral formula for the one-parameter semigroup generated by a sum of contractions of operator-valued tensors. The formula expresses the semigroup in terms of the semigroups generated by the quadratic forms associated with the tensors. It is used to construct solutions of some wave and diffusion equations from solutions of related lower-dimensional equations.
A note on the “hyperbolic” Bochner-Riesz means
Authors: Carbery Anthony
Published: June 2010

We consider the

L
p

(

R

2

)

{L^p}({{\mathbf {R}}^2})

boundedness properties of the Fourier multiplier

m
(

ξ
1

,

ξ
2

)
=
(
1
−

ξ
1
2

ξ
2
2

)
+
α

 for 

α
>
0

m({\xi _1},{\xi _2}) = (1 - \xi _1^2\xi _2^2)_ + ^\alpha {\text { for }}\alpha > 0

. We prove that if

α
⩾

1
2

\alpha \geqslant \frac {1}{2}

, then

m
m

is bounded on

L
p

{L^p}

,

1
>
p
>
∞

1 > p > \infty

, and that if

α
>
0

\alpha > 0

, then

m
m

is bounded on

L
p

{L^p}

,

4
3

⩽
p
⩽
4

\frac {4}{3} \leqslant p \leqslant 4

.
Distribution of alternation points in uniform polynomial approximation
Authors: Lorentz G. G.
Published: June 2010

For a continuous function

f
f

on

[

0
,
1

]

\left [ {0,1} \right ]

, we discuss the points where the polynomial

P
n

(
x
)

{P_n}\left ( x \right )

of best uniform approximation deviates most from

f

(
x
)

f\left ( x \right )

, and the signs of the difference

f

(
x
)

−

P
n

(
x
)

f\left ( x \right ) - {P_n}\left ( x \right )

alternate. We show that these points can be very irregularly distributed in

[

0
,
1

]

\left [ {0,1} \right ]

, even if

f
f

is entire.
Counterexamples to several problems concerning 𝐺_{𝛿}-embeddings
Authors: Ghoussoub N. , Maurey B.
Published: June 2010

Let JT (resp. JH) be the James-tree space (resp. the Hagler space). We observe that two canonical subspaces of

J

T

∗

{\text {J}}{{\text {T}}^ * }

and

J

H

∗

{\text {J}}{{\text {H}}^ * }

,

G
δ

{G_\delta }

-embed in

l
2

{l_2}

even though they fail the Radon-Nikodym property. Such

G
δ

{G_\delta }

-embeddings cannot be the product of a finite number of semi-embeddings. This answers negatively several questions of Bourgain-Rosenthal [1] and Rosenthal [8].