bluuebunny/crossref_metadata_2025_split huggingface dataset:
Concerning diagonal similarity of irreducible matrices
Authors: Hartfiel D. J. 
Published: June 2010

If

A
=
(

a

i
j

)

A = ({a_{ij}})

is an

n
×
n

n \times n

irreducible matrix, then there are positive numbers

d
1

,

d
2

,
⋯

{d_1},{d_2}, \cdots

,

d
n

{d_n}

so that

∑
k

d
i

a

i
k

d
k

−
1

=

∑
k

d
k

a

k
i

d
i

−
1

\sum \nolimits _k {{d_i}{a_{ik}}d_k^{ - 1} = } \sum \nolimits _k {{d_k}{a_{ki}}d_i^{ - 1}}

for each

i
∈
{
1
,
2
,
⋯
,
n
}

i \in \{ 1,2, \cdots ,n\}

. Further, the numbers

d
1

,

d
2

,
⋯

{d_1},{d_2}, \cdots

,

d
n

{d_n}

are unique up to scalar multiples.
On the smoothness of eigenfunctions of hyponormal singular integral operators
Authors: Clancey Kevin 
Published: June 2010

Fix

φ
∈

L
∞

(
E
)

\varphi \in {L^\infty }(E)

and let

E
⊂
R

E \subset R

be bounded and measurable; for

1
>
p
>
∞

1 > p > \infty

consider the bounded linear operator
\[

T
f
(
s
)
=
s
f
(
s
)
+

φ
(
s
)

π

∫
E
∗

φ
¯

(
t
)
f
(
t
)

t
−
s

d
t

a

.e

.

s
∈
E

Tf(s) = sf(s) + \frac {{\varphi (s)}}{\pi }\int _E^\ast \frac {{\bar \varphi (t)f(t)}}{{t - s}}dt\;{\mkern 1mu} {\text {a}}{\text {.e}}{\text {. }}s \in E

\]
where

f
∈

L
p

(
E
)

f \in {L^p}(E)

. If

ν
=
λ
+
i
μ
∈
C

\nu = \lambda + i\mu \in C

then there are no nonzero

L
p

(
E
)

{L^p}(E)

solutions of

T
f
=
ν
f

Tf = \nu f

for

p
>
2

p > 2

in case

λ
\lambda

is a point of positive Lebesgue density in the complement of

E
E

.
On the exponent of the ideal class groups of complex quadratic fields.
Authors: Boyd David W. , Kisilevsky H. 
Published: June 2010

Let

m
(
d
)

m(d)

denote the exponent of the ideal class group of the complex quadratic field

Q
(
√
d
)

Q(\surd d)

, where

d
>
0

d > 0

is a fundamental discriminant. It is shown that there are only finitely many

d
d

for which

m
(
d
)
=
3

m(d) = 3

. Assuming the extended Riemann Hypothesis, it is shown that

m
(
d
)
→
∞

as

|

d

|

→
∞

m(d) \to \infty {\text { as }}|d| \to \infty

.
Positive transformations restricted to subspaces and inequalities among their proper values
Authors: Amir-Moéz A. R. , Perry C. R. 
Published: June 2010

Let A be a positive Hermitian transformation on an n-dimensional unitary space

E
n

{E_n}

with proper values

a
1

≧
⋯
≧

a
n

{a_1} \geqq \cdots \geqq {a_n}

. Let

b
1

≧
⋯
≧

b
k

{b_1} \geqq \cdots \geqq {b_k}

be the proper values of

A

|

M

A|M

, where M is a proper subspace of

E
n

{E_n}

and

c
1

≧
⋯
≧

c
h

{c_1} \geqq \cdots \geqq {c_h}

be the proper values of

A

|

M
⊥

A|{M^ \bot }

. Let

i
1

>
⋯
>

i
r

{i_1} > \cdots > {i_r}

and

j
1

>
⋯
>

j
r

{j_1} > \cdots > {j_r}

be sequences of positive integers, with

i
r

≦
k

{i_r} \leqq k

and

j
r

≦
h

{j_r} \leqq h

. Then

(

b

i
1

⋯

b

i
r

)
⋅
(

c

j
1

⋯

c

j
r

)
≧
(

a

n
−
r
+
1

⋯

a
n

)
(

a

i
1

+

j
1

−
1

⋯

a

i
r

+

j
r

−
1

)

({b_{{i_1}}} \cdots {b_{{i_r}}}) \cdot ({c_{{j_1}}} \cdots {c_{{j_r}}}) \geqq ({a_{n - r + 1}} \cdots {a_n})({a_{{i_1} + {j_1} - 1}} \cdots {a_{{i_r} + {j_r} - 1}})

. In this article generalizations of this inequality have been studied.
An inequality of Turán type for Jacobi polynomials
Authors: Gasper George 
Published: June 2010

For Jacobi polynomials

P
n

(
α
,
β
)

(
x
)
,
α
,
β
>
−
1

P_n^{(\alpha ,\beta )}(x),\alpha ,\beta > - 1

, let
\[

R
n

(
x
)
=

P
n

(
α
,
β
)

(
x
)

P
n

(
α
,
β
)

(
1
)

,

Δ
n

(
x
)
=

R
n
2

(
x
)
−

R

n
−
1

(
x
)

R

n
+
1

(
x
)
.

{R_n}(x) = \frac {{P_n^{(\alpha ,\beta )}(x)}}{{P_n^{(\alpha ,\beta )}(1)}},\quad {\Delta _n}(x) = R_n^2(x) - {R_{n - 1}}(x){R_{n + 1}}(x).

\]
We prove that
\[

Δ
n

(
x
)
≧

(
β
−
α
)
(
1
−
x
)

2
(
n
+
α
+
1
)
(
n
+
β
)

R
n
2

(
x
)
,

−
1
≦
x
≦
1
,
n
≧
1
,

{\Delta _n}(x) \geqq \frac {{(\beta - \alpha )(1 - x)}}{{2(n + \alpha + 1)(n + \beta )}}R_n^2(x),\quad - 1 \leqq x \leqq 1,n \geqq 1,

\]
with equality only for

x
=
±
1

x = \pm 1

. This shows that the Turán inequality

Δ
n

(
α
)
≧
0
,
−
1
≦
x
≦
1

{\Delta _n}(\alpha ) \geqq 0, - 1 \leqq x \leqq 1

, holds if and only if

β
≧
α
>
−
1

\beta \geqq \alpha > - 1

.
A random Fredholm integral equation
Authors: Padgett W. J. , Tsokos Chris P. 
Published: June 2010

The aim of this paper is the study of a random or stochastic integral equation of the Fredholm type given by

x
(
t
;
ω
)
=
h
(
t
;
ω
)
+

∫
0
∞

k
0

(
t
,

τ
;
ω
)
e
(
τ
,
x
(
τ
;
ω
)
)

d
τ
,
t
≧
0

x(t;\omega ) = h(t;\omega ) + \smallint _0^\infty {{k_0}(t,} \tau ;\omega )e(\tau ,x(\tau ;\omega ))\;d\tau , t \geqq 0

, where

ω
∈
Ω

\omega \in \Omega

, the supporting set of the probability measure space

(
Ω
,
A
,
P
)

(\Omega ,A,P)

. The existence and uniqueness of a random solution to the above stochastic integral equation is considered. A random solution,

x
(
t
;
ω
)

x(t;\omega )

, of such a random equation is defined to be a random function which satisfies the equation almost surely. Several theorems and useful special cases are presented which give conditions such that a random solution exists.
A lower bound for the permanent of a (0,1)-matrix
Authors: Gibson P. M. 
Published: June 2010

Let

A
=
(

a

i
j

)

A = ({a_{ij}})

be an n-square fully indecomposable

(
0
,
1
)

(0,1)

-matrix. It is shown that if each row sum of A is at least k then per

A
≧

∑

i
,
j
=
1

n

a

i
j

−
2
n
+
2
+

∑

m
=
1

k
−
1

(
m
!
−
1
)

A \geqq \sum \nolimits _{i,j = 1}^n{a_{ij}} - 2n + 2 + \sum \nolimits _{m = 1}^{k - 1}(m! - 1)

. This improves an inequality obtained by H. Minc.
On a subclass of square integrable martingales
Authors: Isaacson Dean 
Published: June 2010

Let

M

2
∗

\mathcal {M}_2^ \ast

denote the class of continuous, nowhere constant, square integrable martingales,

M
(
t
)
=
X
(

⟨
M

⟩
t

)

M(t) = X({\langle M\rangle _t})

, for which

⟨
M

⟩
t

{\langle M\rangle _t}

is a time change on the

σ
\sigma

-fields generated by the Brownian motion

X
(
t
)

X(t)

. It is shown that if

M
(
t
)
∈

M

2
∗

M(t) \in \mathcal {M}_2^ \ast

, then the family of

σ
\sigma

-fields generated by

M
(
t
)

M(t)

is a right continuous family. If

M
(
t
)
∈

M

2
∗

M(t) \in \mathcal {M}_2^ \ast

and if

σ
{
M
(
s
)
:
s
≦
t
}
=
σ
{
X
(
s
)
:
s
≦
t
}

\sigma \{ M(s):s \leqq t\} = \sigma \{ X(s):s \leqq t\}

for some Brownian motion

X
(
t
)

X(t)

, then

M
(
t
)
=

∫
0
t

Φ
(
s
)
d
X
(
s
)

M(t) = \smallint _0^t {\Phi (s)dX(s)}

and

X
(
t
)
=

∫
0
t

(
1

/

Φ
(
s
)
)
d
M
(
s
)

X(t) = \smallint _0^t {(1/\Phi (s))dM(s)}

for some process

Φ
(
s
)

\Phi (s)

with

Φ
(
s
)
≠
0

\Phi (s) \ne 0

a.e.

d
t
×
d
P

dt \times dP

.
A fixed point theorem for asymptotically nonexpansive mappings
Authors: Goebel K. , Kirk W. A. 
Published: June 2010

Let K be a subset of a Banach space X. A mapping

F
:
K
→
K

F:K \to K

is said to be asymptotically nonexpansive if there exists a sequence

{

k
i

}

\{ {k_i}\}

of real numbers with

k
i

→
1

{k_i} \to 1

as

i
→
∞

i \to \infty

such that

‖

F
i

x
−

F
i

y

‖

≦

k
i

‖

x
−
y

‖

,
x
,
y
∈
K

\left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K

. It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if

F
:
K
→
K

F:K \to K

is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.
Finitely generated submodules of differentiable functions
Authors: Roth B. 
Published: June 2010

Suppose

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

is the Cartesian product of the space of real-valued m-times continuously differentiable functions on an open set

Ω
\Omega

in

R
n

{R^n}

with itself p-times where m is finite and

Ω
\Omega

is connected.

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

is a

E

m

(
Ω
)

{\mathcal {E}^m}(\Omega )

-module. The finitely generated submodules of

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

are

im

(
F
)

{\text {im}}(F)

where

F
:

[

E

m

(
Ω
)

]
q

→

[

E

m

(
Ω
)

]
p

F:{[{\mathcal {E}^m}(\Omega )]^q} \to {[{\mathcal {E}^m}(\Omega )]^p}

is a

p
×
q

p \times q

matrix

(

f

i
j

)

1
≦
i
≦
p
,

1
≦
j
≦
q

,

f

i
j

∈

E

m

(
Ω
)

{({f_{ij}})_{1 \leqq i \leqq p,}}_{1 \leqq j \leqq q},{f_{ij}} \in {\mathcal {E}^m}(\Omega )

. In the present paper, it is shown that

im

(
F
)

{\text {im}}(F)

is closed in

[

E

m

(
Ω
)

]
p

{[{\mathcal {E}^m}(\Omega )]^p}

if and only if the rank of the matrix

(

f

i
j

(
x
)

)

1
≦
i
≦
p
,
1
≦
j
≦
q

{({f_{ij}}(x))_{1 \leqq i \leqq p,1 \leqq j \leqq q}}

is constant for

x
∈
Ω

x \in \Omega

. Applications are made to systems of division problems for distributions.
Maintenance of oscillations under the effect of a periodic forcing term
Authors: Kartsatos Athanassios G. 
Published: June 2010

A necessary and sufficient condition is given for the oscillation of all solutions of the differential equation
\[

x

(
n
)

+
P
(
t
,
x
,

x
′

,
⋯
,

x

(
n
−
1
)

)
=
Q
(
t
)

{x^{(n)}} + P(t,x,x’, \cdots ,{x^{(n - 1)}}) = Q(t)

\]
where

x
1

P
(
t
,

x
1

,

x
2

,
⋯
,

x
n

)
>
0

{x_1}P(t,{x_1},{x_2}, \cdots ,{x_n}) > 0

for every

x
1

≠
0

{x_1} \ne 0

, and Q is a continuous periodic function. This result answers a question recently raised by J. S. W. Wong. It is also shown that a well-known sufficient condition for the existence of at least one nonoscillatory solution of the unperturbed equation guarantees, for a large class of equations, the nonexistence of bounded oscillatory solutions.
On the bordism ring of complex projective space
Authors: Schochet Claude 
Published: June 2010

The bordism ring

M

U
∗

(
C

P
∞

)

M{U_\ast }(C{P^\infty })

is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

are considered, where E is an oriented ring spectrum,

R
=

π
∗

(
E
)

R = {\pi _\ast }(E)

, and

p
R
=
0

pR = 0

for a prime p. It is known that

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

is freely generated as an R-module by elements

{

β
r

|

r
≧
0
}

\{ {\beta _r}|r \geqq 0\}

. The ring structure, however, is not known. It is shown that the elements

{

β

p
r

|

r
≧
0
}

\{ {\beta _{{p^r}}}|r \geqq 0\}

form a simple system of generators for

E
∗

(
C

P
∞

)

{E_\ast }(C{P^\infty })

and that

β

p
r

p

≡

s

p
r

β

p
r

mod
(

β
1

,
⋯
,

β

p

r
−
1

)

\beta _{{p^r}}^p \equiv {s^{{p^r}}}{\beta _{{p^r}}}\bmod ({\beta _1}, \cdots ,{\beta _{{p^{r - 1}}}})

for an element

s
∈
R

s \in R

(which corresponds to

[
C

P

p
−
1

]

[C{P^{p - 1}}]

when

E
=
M
U

Z
p

)

E = MU{Z_p})

. This may lead to information concerning

E
∗

(
K
(
Z
,
n
)
)

{E_\ast }(K(Z,n))

.
Fixed point theorems in reflexive Banach spaces
Authors: Kannan R. 
Published: June 2010

In this paper fixed point theorems are established first for mappings

T
T

, mapping a closed bounded convex subset

K
K

of a reflexive Banach space into itself and satisfying
\[

|

|

T
x
−
T
y

|

|

≦

1
2

{

|

|

x
−
T
x

|

|

+

|

|

y
−
T
y

|

|

,

x
,
y
∈
K
,

||Tx - Ty|| \leqq \tfrac {1}{2}\{ ||x - Tx|| + ||y - Ty||,\quad x,y \in K,

\]
and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.
On (𝑛,𝑛)-zeros of solutions of linear differential equations of order 2𝑛
Authors: Ridenhour Jerry R. 
Published: June 2010

Sufficient conditions on the coefficients

p

2
n

,

p

2
n
−
1

,
⋯
,

p
0

{p_{2n}},{p_{2n - 1}}, \cdots ,{p_0}

are given which guarantee that no nontrivial solution of

p

2
n

y

(
2
n
)

+

p

2
n
−
1

y

(
2
n
−
1
)

+
⋯
+

p
0

y
=
0

{p_{2n}}{y^{(2n)}} + {p_{2n - 1}}{y^{(2n - 1)}} + \cdots + {p_0}y = 0

has two distinct zeros each of order at least

n
n

. These conditions are in the form of

n
n

inequalities which are satisfied by linear combinations of the coefficients and their derivatives.
Sums of quotients of additive functions
Authors: De Koninck Jean-Marie 
Published: June 2010

Denote by

ω
(
n
)

\omega (n)

and

Ω
(
n
)

\Omega (n)

the number of distinct prime factors of

n
n

and the total number of prime factors of

n
n

, respectively. Given any positive integer

α
\alpha

, we prove that
\[

∑

2
≦
n
≦
x

Ω
(
n
)

/

ω

(
n
)
=
x
+
x

∑

i
=
1

α

a
i

/

(
log
⁡
log
⁡
x
)

i

+
O

(
x

/

log
⁡
log
⁡
x
)

α
+
1

)
,

\sum \limits _{2 \leqq n \leqq x} {\Omega (n)/\omega } (n) = x + x\sum \limits _{i = 1}^\alpha {{a_i}/{{(\log \log x)}^i} + O{{(x/\log \log x)}^{\alpha + 1}}),}

\]
where

a
1

=

∑
p

1

/

p
(
p
−
1
)

{a_1} = \sum \nolimits _p {1/p(p - 1)}

and all the other

a
i

{a_i}

’s are computable constants. This improves a previous result of R. L. Duncan.
Problem 26 of L. Fuchs
Authors: Hsü Chin Shui 
Published: June 2010

This solves the following problem: Which Abelian groups are the inverse limits of Abelian groups, each of which is a finite direct sum of quasi-cyclic and bounded Abelian groups? (Here quasi-cyclic means isomorphic to some

Z
(

p
∞

)

Z({p^\infty })

.) A necessary and sufficient condition for an Abelian group to be such is that it takes the form

A
r

⊕

Π
p

Hom
z

⁡
(

A
p

,
Z
(

p
∞

)
)

{A_r} \oplus \Pi _p\operatorname {Hom}_z({A_p},Z({p^\infty }))

where

A
r

{A_r}

is complete and reduced, the

A
p

{A_p}

are torsion-free and the direct product is taken over the set of prime numbers.
On restricted weak type (1,1)
Authors: Moon K. H. 
Published: June 2010

Let

{

S
k

}

k
≧
1

{\{ {S_k}\} _{k \geqq 1}}

be a sequence of linear operators defined on

L
1

(

R
n

)

{L^1}({R^n})

such that for every

f
∈

L
1

(

R
n

)
,

S
k

f
=
f
∗

g
k

f \in {L^1}({R^n}),{S_k}f = f \ast {g_k}

for some

g
k

∈

L
1

(

R
n

)
,
k
=
1
,
2
,
⋯

{g_k} \in {L^1}({R^n}),k = 1,2, \cdots

, and

T
f
(
x
)
=

sup

k
≧
1

|

S
k

f
(
x
)

|

Tf(x) = {\sup _{k \geqq 1}}|{S_k}f(x)|

. Then the inequality

m
{
x
∈

R
n

;
T
f
(
x
)
>
y
}
≦
C

y

−
1

∫

R
n

|

f
(
t
)

|

d
t

m\{ x \in {R^n};Tf(x) > y\} \leqq C{y^{ - 1}}\smallint _{{R^n}} {|f(t)|dt}

holds for characteristic functions f (T is of restricted weak type (1, 1)) if and only if it holds for all functions

f
∈

L
1

(

R
n

)

f \in {L^1}({R^n})

(T is of weak type (1, 1)). In particular, if

S
k

f

{S_k}f

is the kth partial sum of Fourier series of f, this theorem implies that the maximal operator T related to

S
k

{S_k}

is not of restricted weak type (1, 1).
An inequality for analytic functions
Authors: Kamowitz Herbert 
Published: June 2010

If

F
F

denotes the boundary value of a function

f
∈

H
p

,
1
≤
p
≤
∞

f \in {H^p}, 1 \leq p \leq \infty

, the infimum of the measure of

{
θ

|

|

F
(
θ
)

|

>
A
}

\{ \theta |\;|F(\theta )| > A\}

for given

A
,
0
>
A
>

|

f
(
0
)

|

,

|

|

f

|

|

H
p

=
1

A, 0 > A > |f(0)|, ||f|{|_{{H^p}}} = 1

, is determined.
A characterization of parallel ovaloids
Authors: Koutroufiotis Dimitri 
Published: June 2010

Two ovaloids

S
,

S
¯

S,\bar S

can be mapped diffeomorphically onto each other by equal inner normals. If, under this mapping, principal directions are preserved and
\[

[
(
p
−

p
¯

)
−
(
c

k
1

−
1

−

k
¯

1

−
1

)
]
[
(
p
−

p
¯

)
−
(
c

k
2

−
1

−

k
¯

2

−
1

)
]
≤
0

[(p - \bar p) - (ck_1^{ - 1} - \bar k_1^{ - 1})][(p - \bar p) - (ck_2^{ - 1} - \bar k_2^{ - 1})] \leq 0

\]
everywhere on the unit sphere for a certain constant

c
c

, then

c
=
1

c = 1

and

p
−

p
¯

p - \bar p

= constant. Here

p
,

p
¯

p,\bar p

are the support functions,

k
1

{k_1}

, and

k
2

{k_2}

the principal curvatures of

S
,

k
¯

1

S,{\bar k_1}

and

k
¯

2

{\bar k_2}

the corresponding principal curvatures of

S
¯

\bar S

. Various characterizations of the sphere are obtained as corollaries.
On the upper bound of the number of real roots of a random algebraic equation with infinite variance. II
Authors: Samal G. , Mishra M. N. 
Published: June 2010

Let

N
n

{N_n}

be the number of real roots of

∑

v
=
0

n

a
v

ξ
v

x
v

=
0

\sum \nolimits _{v = 0}^n {{a_v}{\xi _v}{x^v}} = 0

where

ξ
v

{\xi _v}

’s are independent random variables identically distributed with a common characteristic function

exp
⁡
(
−
C

|

t

|

α

)
;
C

\exp ( - C|t{|^\alpha });C

is a positive constant,

a
0

,

a
1

,
⋯
,

a
n

{a_0},{a_1}, \cdots ,{a_n}

are nonzero real numbers such that

k
n

=

max

0
≦
v
≦
n

|

a
v

|

=
O
(

n
β

/

log
⁡
n
)

{k_n} = {\max _{0 \leqq v \leqq n}}|{a_v}| = O({n^\beta }/\log n)

. Then (i)

Pr
{

Sup

n
>

n
0

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n
0

3
α
−
2
−
β

,
1
≦
α
≦
2
,
0
>
β
>
1

\Pr \{ {\operatorname {Sup} _{n > {n_0}}}{N_n}/{(\log n)^2} > \mu \} > \mu ’/n_0^{3\alpha - 2 - \beta },1 \leqq \alpha \leqq 2,0 > \beta > 1

; (ii)

Pr
{

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n
,
α
≧
1

\Pr \{ {N_n}/{(\log n)^2} > \mu \} > \mu ’/n,\alpha \geqq 1

; (iii)

Pr
{

N
n

/

(
log
⁡
n

)
2

>
μ
}
>

μ
′

/

n

3
α
−
1
−
β

,
1
≦
α
≦
2

\Pr \{ {N_n}/{(\log n)^2} > \mu \} > \mu ’/{n^{3\alpha - 1 - \beta }},1 \leqq \alpha \leqq 2

.
Almost everywhere convergence for sequences of continuous functions
Authors: Schrader K. , Umamaheswaram S. 
Published: June 2010

The main result in this paper is the following theorem. Theorem 1.1. Let

{
y
k
}
,
y
k
:
I
→
R

\{ yk\} ,yk:I \to R

, be a sequence of continuous real valued functions defined on the bounded interval

I
I

. Let

D

k
j

=
{
x
:
x
∈
I
,

|

y
k
(
x
)
−
y
j
(
x
)

|

>
0
}
=

∪
n

I

k
j
n

{D_{kj}} = \{ x:x \in I,|yk(x) - yj(x)| > 0\} = { \cup _n}{I_{kjn}}

, where each

I

k
j
n

{I_{kjn}}

is a relatively open subinterval of

I
I

and

I

k
j
n

∩

I

k
j
m

=
ϕ

{I_{kjn}} \cap {I_{kjm}} = \phi

for

n
≠
m

n \ne m

. Assume there exists a function

ϕ
,
ϕ
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)

\phi ,\phi :(0, + \infty ) \to (0, + \infty )

, such that

lim

r
→

0
+

(
ϕ
(
r
)

/

r
)
=
+
∞
,
ϕ

{\lim _{r \to {0^ + }}}(\phi (r)/r) = + \infty ,\phi

is a nondecreasing function of

r
r

and
\[

∑
n

ϕ
(
μ
(

I

k
j
n

)
)

sup

x
∈

I

k
j
n

|

y
k

(
x
)
−

y
j

(
x
)

|

p

≤
M

\sum \limits _n {\phi (\mu ({I_{kjn}}))\sup \limits _{x\in {I_{kjn}}} |{y_k}(x) - {y_j}(x){|^p} \leq M}

\]
for all

k
,
j

k,j

sufficiently large, where

μ
\mu

is Lebesgue measure and

0
>
p
>
+
∞

0 > p > + \infty

. Then

{

y
k

}

\{ {y_k}\}

contains a subsequence which converges almost everywhere to a Lebesgue measurable function

y
y

.
On Euler’s criterion for cubic nonresidues
Authors: Williams Kenneth S. 
Published: June 2010

If

p
p

is a prime

≡
1

(
mod

3
)

\equiv 1 \pmod 3

there are integers

L
L

and

M
M

such that

4
p
=

L
2

+
27

M
2

,
L
≡
1

(
mod

3
)

4p = {L^2} + 27{M^2},L \equiv 1\pmod 3

. Indeed

L
L

and

M
2

{M^2}

are unique. If

D
D

is a cubic nonresidue

(
mod

p
)

\pmod p

it is shown how to choose the sign of

M
M

so that
\[

D

(
p
−
1
)

/

3

≡
(
L
+
9
M
)

/

(
L
−
9
M
)

(
mod

p
)
.

{D^{(p - 1)/3}} \equiv (L + 9M)/(L - 9M)\pmod p.

\]
The case

D
=
2

D = 2

has been treated by Emma Lehmer.
Inequalities in dimension theory for posets
Authors: Trotter William T. 
Published: June 2010

The dimension of a poset

(
X
,
P
)

(X,P)

, denoted

dim
⁡
(
X
,
P
)

\dim (X,P)

, is the minimum number of linear extensions of

P
P

whose intersection is

P
P

. It follows from Dilworth’s decomposition theorem that

dim
⁡
(
X
,
P
)
≤
width
⁡
(
X
,
P
)

\dim (X,P) \leq \operatorname {width} (X,P)

. Hiraguchi showed that

dim
⁡
(
X
,
P
)
≤

|

X

|

/

2

\dim (X,P) \leq |X|/2

. In this paper,

A
A

denotes an antichain of

(
X
,
P
)

(X,P)

and

E
E

the set of maximal elements. We then prove that

dim
⁡
(
X
,
P
)
≤

|

X
−
A

|

;
dim
⁡
(
X
,
P
)
≤
1
+
width
⁡
(
X
−
E
)
;

\dim (X,P) \leq |X - A|;\dim (X,P) \leq 1 + \operatorname {width} (X - E);

and

dim
⁡
(
X
,
P
)
≤
1
+
2
width
⁡
(
X
−
A
)

\dim (X,P) \leq 1 + 2\operatorname {width} (X - A)

. We also construct examples to show that these inequalities are sharp.
On ergodic sequences of measures
Authors: Blum J. R. , Cogburn R. 
Published: June 2010

Let

Z
Z

be the group of integers and

Z
¯

\bar Z

its Bohr compactification. A sequence of probability measures

{

μ
n

,
n
=
1
,
2
,
…
}

\{ {\mu _n},n = 1,2, \ldots \}

defined on

Z
Z

is said to be ergodic provided

μ
n

{\mu _n}

converges weakly to

μ
¯

\bar \mu

, the Haar measure on

Z
¯

\bar Z

. Let

A
n

⊂
Z
,
n
=
1
,
2
,
…

{A_n} \subset Z,n = 1,2, \ldots

and define

μ
n

{\mu _n}

by

μ
n

(
B
)
=

|

A
n

∩
B

|

/

|

A
n

|

{\mu _n}(B) = |{A_n} \cap B|/|{A_n}|

where

|

B

|

|B|

is the cardinality of

B
B

. Then it is easy to show that if

|

A
n

∩

A
n

+
k

|

/

|

A
n

|

→
1

|{A_n} \cap {A_n} + k|/|{A_n}| \to 1

for every

k
∈
Z

k \in Z

, then

μ
n

{\mu _n}

is ergodic. Let

0
≤

p
k

≤
1

0 \leq {p_k} \leq 1

. In this paper we construct (random) sequences

{

μ
n

}

\{ {\mu _n}\}

which are ergodic, and such that

lim
(

|

A
n

∩

A
n

+
k

|

/

|

A
n

|

)
=

p
k

\lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k}

, for every

k
∈
Z

k \in Z

.
On values of exponential sums
Authors: An Chungming 
Published: June 2010

An exponential sum is defined by
\[

G
(
F
,
φ
,
α
)
=

∑

γ
ϵ

(
Z

/

q
Z
)

n

exp
⁡
(
2
π
i
(
φ
F
(
γ
)
+
⟨
α
,
γ
⟩

)
)

G(F,\varphi ,\alpha ) = \sum \limits _{\gamma \epsilon {{(Z/qZ)}^n}} {\exp (2\pi i(\varphi F(\gamma ) + \langle \alpha ,\gamma \rangle } ))

\]
for

φ
=
a

/

q
ϵ
Q
,
α
ϵ

R
n

\varphi = a/q\epsilon Q,\alpha \epsilon {R^n}

, and a positive-definite form

F
(
x
)

F(x)

in

n
n

variables of degree

δ
\delta

. Its value is studied and the definition is extended to an irrational

φ
\varphi

.
On perturbing bases of complex exponentials in 𝐿² (-𝜋,𝜋)
Authors: Young Robert M. 
Published: June 2010

A sequence of complex exponentials

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is said to be a Riesz basis for

L
2

(
−
π
,

π
)

{L^2}( - \pi ,\;\pi )

if each function in the space has a unique representation

f
=
Σ

c
n

e

i

λ
n

t

f = \Sigma {c_n}{e^{i{\lambda _n}t}}

, with

A
Σ

|

c
n

|

2

⩽

|

|

f

|

|

2

⩽
B
Σ

|

c
n

|

2

A\Sigma |{c_n}{|^2} \leqslant ||f|{|^2} \leqslant B\Sigma |{c_n}{|^2}

. It is known, for example, that if

|

λ
n

−
n

|

⩽
L
>
1

/

4
(
−
∞
>
n
>
∞
)

|{\lambda _n} - n| \leqslant L > 1/4( - \infty > n > \infty )

, then

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a Riesz basis. In this note we show that not only the orthonormal basis

{

e

i
n
t

}

\{ {e^{int}}\}

, but any Riesz basis of complex exponentials can be suitably perturbed.
Factors and roots of the van der Pol polynomials
Authors: Howard F. T. 
Published: June 2010

The van der Pol polynomials

V
n

(
a
)

{V_n}(a)

are defined by means of
\[

x
3

e

x
a

[
6
x
(

e
x

+
1
)
−
12
(

e
x

−
1
)

]

−
1

=

∑

n
=
0

∞

V
n

(
a
)

x
n

/

n
!
.

{x^3}{e^{xa}}{[6x({e^x} + 1) - 12({e^x} - 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{V_n}} (a){x^n}/n!.

\]
In this paper new properties of these polynomials are derived. It is shown that neither

V

2
n

(
a
)

{V_{2n}}(a)

nor

V

2
n
+
1

(
a
)

/

(
a
−
1

/

2
)

{V_{2n + 1}}(a)/(a - 1/2)

has rational roots, and that if

n
=
2
∙

3
m

,
m
⩾
0

n = 2 \bullet {3^m},m \geqslant 0

, or

n
=

3
m

+

3
t

,
m
>
t
>
0

n = {3^m} + {3^t},m > t > 0

, or

n
=
m
(
p
−
3
)
,
p

n = m(p - 3),p

a prime number,

3
m
>
p

3m > p

, then

V
n

(
a
)

{V_n}(a)

and

V

n
+
1

(
a
)

/

(
a
−
1

/

2
)

{V_{n + 1}}(a)/(a - 1/2)

are both irreducible over the rational field. It is also shown that if

n
=

2
k

n = {2^k}

, then

V
n

(
a
)

{V_n}(a)

is irreducible over the rational field. Finally, possible factors of the van der Pol polynomials are discussed.