bluuebunny/crossref_metadata_2025_split huggingface dataset:
A characterization of M. W. Wilson’s criterion for nonnegative expansions of orthogonal polynomials
Authors: Micchelli Charles A. 
Published: June 2010

Given a nonnegative function

f
(
x
)

f(x)

, M. W. Wilson observed that if

(

1

)

∫
0
∞

f
(
x
)

Q
i

(
x
)

Q
j

(
x
)
d
μ
(
x
)
⩽
0
,

i
≠
j
,

\begin{equation}\int _0^\infty f(x) Q_i(x) Q_j(x)d\mu (x) \leqslant 0,\quad i \ne j, \quad \tag {$1$}\end{equation}

then the polynomials

P
n

(
x
)
,

P
n

(
0
)
=
1

{P_n}(x),{P_n}(0) = 1

, orthogonal relative to

f
(
x
)
d
μ
(
x
)

f(x)d\mu (x)

, have an expansion
\[

P
n

(
x
)
=

∑

k
=
0

n

a

k
n

Q
k

(
x
)

{P_n}(x) = \sum \limits _{k = 0}^n {{a_{kn}}{Q_k}(x)}

\]
with nonnegative coefficients

a

k
n

⩾
0

{a_{kn}} \geqslant 0

where

Q
n

(
x
)
,

Q
n

(
0
)
=
1

{Q_n}(x),{Q_n}(0) = 1

, are orthogonal relative to

d
μ
(
x
)

d\mu (x)

. Recently it was shown that (1) holds for

f
(
x
)
=

x
c

,
0
>
c
>
1

f(x) = {x^c},0 > c > 1

. In this paper we characterize those functions

f
(
x
)

f(x)

for which (1) is valid for all positive measures

d
μ
(
x
)

d\mu (x)

.
Some spaces related to topological inequalities proven by the Erdős-Rado theorem
Authors: Fleissner William G. 
Published: June 2010

The Erdös-Rado theorem is very useful in proving cardinal inequalities in topology. It has been suggested that certain of these inequalities might be strengthened. We note that trees constructed by Jensen and Gregory using various extra axioms of set theory yield several counterexamples to these suggestions; for example, a space

X
,

|

X

|

=

ω
2

,
c
(
X
)
=

ω
1

,
χ
(
X
)
=
ω

X,|X| = {\omega _2},c(X) = {\omega _1},\chi (X) = \omega

, answering a question of Hajnal and Juhász. We consider the apparently similar relation between

|

X

|

,
e
(
X
)

|X|,e(X)

, and

d
Ψ
(
X
)

d\Psi (X)

of Ginsburg and Woods. Using combinatorial consequences of

V
=
L

V = L

, we construct

G
δ

{G_\delta }

tree families, and establish that, assuming

V
=
L

V = L

, an infinite cardinal

κ
\kappa

is weakly compact

iff

d
Ψ
(
X
)
>
κ
,

e
a

(
X
)
⊂
κ

imply

|

X

|

>
κ

{\text {iff}}\,d\Psi (X) > \kappa ,{e_a}(X) \subset \kappa {\text {imply}}|X| > \kappa

. We consider products of countable chain condition spaces, and show that, using Cohen forcing that (

2
ω

{2^\omega }

can be anything allowed by König’s theorem and there are spaces

X
,
Y
,
c
(
X
)
=
c
(
Y
)
=
ω
,
c
(
X
×
Y
)
=

2
ω

X,Y,c(X) = c(Y) = \omega ,c(X \times Y) = {2^\omega }

). A variation is a space W with the property

c
(

W
n

)
=

ω

n
−
1

c({W^n}) = {\omega _{n - 1}}

.
An octic reciprocity law of Scholz type
Authors: Buell Duncan A. , Williams Kenneth S. 
Published: June 2010

The authors [3] have conjectured that if p and q are distinct primes satisfying
\[

p
≡
q
≡
1

(
mod

8
)
,

(
p

/

q

)
4

=

(
q

/

p

)
4

=
+
1
,

p \equiv q \equiv 1\quad \pmod 8,\quad {(p/q)_4} = {(q/p)_4} = + 1,

\]
then
\[

(

p
q

)

8

(

q
p

)

8

=

{

(

ε
p

q

)

4

(

ε
q

p

)

4

,

if

N
(

ε

p
q

)
=
−
1
,

(
−
1
)

h
(
p
q
)

/

4

(

ε
p

q

)

4

(

ε
q

p

)

4

,

if

N
(

ε

p
q

)
=
+
1
,

{\left ( {\frac {p}{q}} \right )_8}{\left ( {\frac {q}{p}} \right )_8} = \left \{ {\begin {array}{*{20}{c}} {{{\left ( {\frac {{{\varepsilon _p}}}{q}} \right )}_4}{{\left ( {\frac {{{\varepsilon _q}}}{p}} \right )}_4},\quad {\text {if}}\;N({\varepsilon _{pq}}) = - 1,} \hfill \\ {{{( - 1)}^{h(pq)/4}}{{\left ( {\frac {{{\varepsilon _p}}}{q}} \right )}_4}{{\left ( {\frac {{{\varepsilon _q}}}{p}} \right )}_4},\quad {\text {if}}\;N({\varepsilon _{pq}}) = + 1,} \hfill \\ \end {array} } \right .

\]
where

ε
p

{\varepsilon _p}

is the fundamental unit of

Q
(

p

)
,
N
(

ε

p
q

)

Q(\sqrt p ),N({\varepsilon _{pq}})

denotes the norm of the unit

ε

p
q

{\varepsilon _{pq}}

, and

h
(
p
q
)

h(pq)

is the class number of

Q
(

p
q

)

Q(\sqrt {pq} )

. A proof of this conjecture is given, which makes use of results of Bucher [2].
An improved estimate for certain Diophantine inequalities
Authors: Liu Ming Chit , Ng Shu Ming , Tsang Kai Man 
Published: June 2010

Let

λ
1

,
…
,

λ
8

{\lambda _1}, \ldots ,{\lambda _8}

be any nonzero real numbers such that not all

λ
j

{\lambda _j}

are of the same sign and not all ratios

λ
j

/

λ
k

{\lambda _j}/{\lambda _k}

are rational. If

η
,
α

\eta ,\alpha

are any real numbers with

0
>
α
>
3

/

70

0 > \alpha > 3/70

then

|

η
+

Σ

j
=
1

8

λ
j

n
j
3

|

>

(
max

n
j

)

−
α

|\eta + \Sigma _{j = 1}^8{\lambda _j}n_j^3| > {(\max {n_j})^{ - \alpha }}

has infinitely many solutions in positive integers

n
j

{n_j}

.
The Ceva property characterizes real, strictly convex Banach spaces
Authors: Valentine J. E. , Wayment S. G. 
Published: June 2010

If a, b, c are distinct collinear points of a metric space, then
\[

(
a
,
b
,
c
)
=

{

a
b

/

b
c

a
m
p
;

if

b

is

between

a

and

c
,

−
(
a
b

/

b
c
)

a
m
p
;

otherwise

.

(a,b,c) = \left \{ {\begin {array}{*{20}{c}} {ab/bc} \hfill & {{\text {if}}\;b\;{\text {is}}\;{\text {between}}\;a\;{\text {and}}\;c,} \hfill \\ { - (ab/bc)} \hfill & {{\text {otherwise}}.} \hfill \\ \end {array} } \right .

\]
A metric space satisfies the Ceva Property provided for each triple of noncollinear points p, q, r, if s,t,u are points distinct from p, q, r on the metric lines

L
(
p
,
q
)
,
L
(
q
,
r
)

L(p,q),L(q,r)

, and

L
(
r
,
p
)

L(r,p)

, respectively, then the metric lines

L
(
p
,
t
)
,
L
(
q
,
u
)

L(p,t),L(q,u)

, and

L
(
r
,
s
)

L(r,s)

have a common point if and only if

(
p
,
s
,
q
)
(
q
,
t
,
r
)
(
r
,
u
,
p
)
=
1

(p,s,q)(q,t,r)(r,u,p) = 1

, and

p
q

/

p
s
≠
p
u

/

p
r

pq/ps \ne pu/pr

. In the euclidean plane, the requirement that

p
q

/

p
s
≠
p
u

/

p
r

pq/ps \ne pu/pr

forces the lines

L
(
r
,
s
)

L(r,s)

and

L
(
q
,
u
)

L(q,u)

to have a common point. Thus the case of parallel lines is avoided and the Ceva Property is meaningful in an arbitrary metric space. The main result of the paper is that a complete, convex, externally convex, metric space is a strictly convex Banach space over the reals if and only if it satisfies the Ceva Property.
Chain of prime ideals in formal power series rings
Authors: de S. Doering Ada Maria , Lequain Yves 
Published: June 2010

Let

R
R

be a Noetherian domain and

P
P

a prime ideal of

R
R

. Then

R
p

[
[

X
1

,
…
,

X
n

]
]

{R_p}[[{X_1}, \ldots ,{X_n}]]

has a maximal chain of prime ideals of length

r
r

if and only if

R

[
[

X
1

,
…
,

X
n

]

]

(
P
,

X
1

,
…
,

X
n

)

R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}

does, if and only if

R

[
[

X
1

,
…
,

X
n

]

]

(
P
,

X
1

,
…
,

X
n

)

R{[[{X_1}, \ldots ,{X_n}]]_{(P,{X_1}, \ldots ,{X_n})}}

does.
On the dimensions of projections of compact subsets of 𝑅^{𝑚}
Authors: Sternfeld Y. 
Published: June 2010

In this note we continue the study of a problem considered earlier by G. Nöbling, S. Mardešić, C. Pixley and the author. Let

W
W

be an

n
n

-dimensional compact subset of

R

m

{{\mathbf {R}}^m}

, and let

{

i
1

,

i
2

,
…
,

i
k

}
⊂
{
1
,
2
,
…
,
m
}

\{ {i_1},{i_2}, \ldots ,{i_k}\} \subset \{ 1,2, \ldots ,m\}

. It is shown that if the projection

P

i
1
,
i
2
,
…
,
i
k

{P_{i1,i2, \ldots ,ik}}

satisfies a certain condition called normality on

W
W

, then there exist

i

k
+
1

{i_{k + 1}}

,

i

k
+
2

,
…
,

i
n

{i_{k + 2}}, \ldots ,{i_n}

so that

dim
⁡

P

i
1
,
i
2
,
…
,

i
k

,

i

k
+
1

,
…
,

i
n

(
W
)
=
n

\dim {P_{i1,i2, \ldots ,{i_k},{i_{k + 1}}, \ldots ,{i_n}}}(W) = n

. It is also shown that at least

(
k
n

)

(_k^n)

such normal projections do exist (for

1
⩽
k
⩽
n

1 \leqslant k \leqslant n

), and an example is constructed to show that the normality condition is not necessary.
On the hyperbolic Riesz means
Authors: El Kohen A. 
Published: June 2010

We define the hyperbolic Riesz means in

R

2

{{\mathbf {R}}^2}

by

H
λ

f
=

(

m
λ

f
^

)
υ

{H_\lambda }f = {({m_\lambda }\hat f)^\upsilon }

where

m
λ

(

ξ
1

,

ξ
2

)
=
(
1
−

(

ξ
1

ξ
2

)
2

)
+
λ

,
λ
⩾
0

{m_\lambda }({\xi _1},{\xi _2}) = (1 - {({\xi _1}{\xi _2})^2})_ + ^\lambda ,\lambda \geqslant 0

, and show that

H
λ

{H_\lambda }

is bounded on

L
p

(

R

2

)

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

for

4
3

⩽
p
⩽
4

\tfrac {4}{3} \leqslant p \leqslant 4

and

λ
>

1
2

\lambda > \tfrac {1}{2}

or

1
>
p
>
∞

1 > p > \infty

and

λ
⩾
1

\lambda \geqslant 1

.
An improved estimate in the method of freezing
Authors: Vinograd Robert E. 
Published: June 2010

Let

x
˙

=
A
(
t
)
x

\dot x = A(t)x

and

λ
k

(
t
)

{\lambda _k}(t)

be the eigenvalues of the matrix

A
(
t
)

A(t)

. The main result of the Method of Freezing [1] states that if

‖

A
(
t
)

‖

⩽
a

\left \| {A(t)} \right \| \leqslant a

. Re

λ
k

(
t
)
⩽

λ
0

{\lambda _k}(t) \leqslant {\lambda _0}

and

‖

A
(
t
)
−
A
(
s
)

‖

⩽
δ

|

t
−
s

|

\left \| {A(t) - A(s)} \right \| \leqslant \delta \left | {t - s} \right |

, then
\[

‖

x
(
t
)

‖

⩽

‖

x
(

t
0

)

‖

D
δ

exp
⁡
(

λ
0

+
2
a

λ
δ

)
(
t
−

t
0

)

(
t
⩾

t
0

)

\left \| {x(t)} \right \| \leqslant \left \| {x({t_0})} \right \|{D_\delta }\exp ({\lambda _0} + 2a{\lambda _\delta })(t - {t_0})\quad (t \geqslant {t_0})

\]
for all solutions of the system, where
\[

λ
δ

=

(

C
n

⋅
δ

/

4

a
2

)

1

/

(
n
+
1
)

{\lambda _\delta } = {({C_n} \cdot \delta /4{a^2})^{1/(n + 1)}}

\]
. The previous best known value.

C
n

=
n
(
n
+
1
)

/

2

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

, is reduced to the substantially smaller value

2

n
n

e

−
n

/

(
n
−
1
)
!
>

2
n

/

π

2{n^n}{e^{ - n}}/(n - 1)! > \sqrt {2n/\pi }

.
Note on rotation set
Authors: Ito Ryuichi 
Published: June 2010

Let

f
f

be an endomorphism of the circle of degree 1 and

f
¯

\bar f

be a lifting of

f
f

. We characterize the rotation set

ρ
(

f
¯

)

\rho (\bar f)

by the set of probability measures on the circle, and prove that if

ρ
+

(

f
¯

)

(

ρ
−

(

f
¯

)
)

{\rho _ + }(\bar f)\;({\rho _ - }(\bar f))

, the upper (lower) endpoint of

ρ
(

f
¯

)

\rho (\bar f)

, is irrational, then

ρ
+

(

R
θ

f
¯

)
>

ρ
+

(

f
¯

)

(

ρ
−

(

R
θ

f
¯

)
>

ρ
−

(

f
¯

)
)

{\rho _ + }({R_\theta }\bar f) > {\rho _ + }(\bar f)\;({\rho _ - }({R_\theta }\bar f) > {\rho _ - }(\bar f))

for any

θ
>
0

\theta > 0

, where

R
θ

(
x
)
=
x
+
θ

{R_\theta }(x) = x + \theta

. As a corollary, if

f
f

is structurally stable, then both

ρ
+

(

f
¯

)

{\rho _ + }(\bar f)

and

ρ
−

(

f
¯

)

{\rho _ - }(\bar f)

are rational.