bluuebunny/crossref_metadata_2025_split huggingface dataset:
The analytic properties of 𝐺_{2𝑛} spaces
Authors: Koehler Donald O. 
Published: June 2010

A complex vector space X will be called an

F

2
n

{F_{2n}}

space if and only if there is a mapping

⟨
⋅
,
⋯
,
⋅
⟩

\langle \cdot , \cdots , \cdot \rangle

from

X

2
n

{X^{2n}}

into the complex numbers such that:

⟨
x
,
⋯
,
x
⟩
>
0

\langle x, \cdots ,x\rangle > 0

if

x
≠
0
;

x
k

→
⟨

x
1

,
⋯
,

x

2
n

⟩

x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle

is linear for

k
=
1
,
⋯
,
n
;
⟨

x
1

,
⋯
,

x

2
n

⟩
=

⟨

x

2
n

,
⋯
,

x
1

⟩
−

k = 1, \cdots ,n;\langle {x_1}, \cdots ,{x_{2n}}\rangle = {\langle {x_{2n}}, \cdots ,{x_1}\rangle ^ - }

where denotes complex conjugate;

⟨

x

σ
(
1
)

,
⋯
,

x

σ
(
n
)

,

y

τ
(
1
)

,
⋯
,

y

τ
(
n
)

⟩
=
⟨

x
1

,
⋯
,

x
n

,

y
1

,
⋯
,

y
n

⟩

\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle

for all permutations

σ
,
τ

\sigma ,\tau

of

{
1
,
⋯
,
n
}

\{ 1, \cdots ,n\}

. In the case of a real vector space the mapping is assumed to be into the reals such that:

⟨
x
,
⋯
,
x
⟩
>
0

\langle x, \cdots ,x\rangle > 0

if

x
≠
0
;

x
k

→
⟨

x
1

,
⋯
,

x

2
n

⟩

x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle

is linear for

k
=
1
,
⋯
,
2
n
;
⟨

x

σ
(
1
)

,
⋯
,

x

σ
(
2
n
)

⟩
=
⟨

x
1

,
⋯
,

x

2
n

⟩

k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle

for all permutations

σ
\sigma

of

{
1
,
⋯
,
2
n
}

\{ 1, \cdots ,2n\}

. In either case, if

‖
x
‖

=

⟨
x
,
⋯
,
x

⟩

1

/

2
n

\left \| x \right \| = {\langle x, \cdots ,x\rangle ^{1/2n}}

defines a norm, X is called a

G

2
n

{G_{2n}}

space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an

F

2
n

{F_{2n}}

space is a

G

2
n

{G_{2n}}

space if and only if

|

⟨
x
,
y
,
⋯
,
y
⟩

|

2
n

≦
⟨
x
,
⋯
,
x
⟩

⟨
y
,
⋯
,
y

⟩

2
n
−
1

|\langle x,y, \cdots ,y\rangle {|^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}}

and that

G

2
n

{G_{2n}}

spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).
The solution of a nonlinear Gronwall inequality
Authors: Helton Burrell W. 
Published: June 2010

This paper extends some of the earlier results of J. V. Herod, W. W. Schmaedeke and G. R. Sell, and B. W. Helton and shows that, under the given conditions, (1) there is a function

u
u

satisfying the inequality
\[

f
(
x
)
≦
h
(
x
)
+
(
R
L
)

∫
a
x

(
f
G
+
f
H
)

f(x) \leqq h(x) + (RL)\int _a^x {(fG + fH)}

\]
such that, if

f
f

satisfies the given inequality, then

f
(
x
)
≦
u
(
x
)

f(x) \leqq u(x)

; and (2) there is a function

u
u

satisfying the inequality
\[

0
>
f
(
x
)
≦
k
+
(
R
L
)

∫
a
x

[
(
f

G
1

+

f

p
n

G
2

)
+
(
f

H
1

+

f

p
n

H
2

)
]
,

0 > f(x) \leqq k + (RL)\int _a^x {[(f{G_1} + {f^{pn}}{G_2}) + (f{H_1} + {f^{pn}}{H_2})],}

\]
where

n
n

is a positive integer and

p
=
±
1

p = \pm 1

and

p
n
≠
1

pn \ne 1

, such that, if

f
f

satisfies the given inequality, then

f
(
x
)
≦
u
(
x
)

f(x) \leqq u(x)

.
Distortions properties of alpha-starlike functions
Authors: Miller Sanford S. 
Published: June 2010

Let

α
\alpha

be real and suppose that

f
(
z
)
=
z
+

Σ
2
∞

a
n

z
n

f(z) = z + \Sigma _2^\infty {a_n}{z^n}

is regular in the unit disc

D
D

with

f
(
z
)

f
′

(
z
)
≠
0

f(z)f’(z) \ne 0

in

0
>

|

z

|

>
1

0 > |z| > 1

. If

Re
⁡
[
(
1
−
α
)
z

f
′

(
z
)

/

f
(
z
)
+
α
(
(
z
f
(
z
)

/

f
′

(
z
)
)
+
1
)
]
>
0

\operatorname {Re} [(1 - \alpha )zf’(z)/f(z) + \alpha ((zf(z)/f’(z)) + 1)] > 0

for

z
∈
D

z \in D

, then

f
(
z
)

f(z)

is said to be an alpha-starlike function. These functions are univalent and they very naturally unify the classes of starlike

(
α
=
0
)

(\alpha = 0)

and convex

(
α
=
1
)

(\alpha = 1)

functions. The author obtains the

1
4

\tfrac {1}{4}

-theorem, sharp bounds on

|

f
(
z
)

|

|f(z)|

and

|

f
′

(
z
)

|

|f’(z)|

, and growth conditions on

M
(
r
)

M(r)

.
The homotopy groups of spaces whose cohomology is a 𝑍_{𝑝} truncated polynomial algebra
Authors: Shar Albert 
Published: June 2010

On such a space one can define a “Hopf” invariant homomorphism

h
:

π

q
n
−
1

(
K
)
→
Z

h:{\pi _{qn - 1}}(K) \to Z

in two ways. We prove both definitions are equivalent and show that

p

π
i

(
K
)
≃

p

π

i
−
1

(

S

n
−
1

)
⊕

p

π
i

(

S

q
n
−
1

)

{}_p{\pi _i}(K) \simeq {}_p{\pi _{i - 1}}({S^{n - 1}}) \oplus {}_p{\pi _i}({S^{qn - 1}})

if and only if there is an

α
∈

π

q
n
−
1

(
K
)

\alpha \in {\pi _{qn - 1}}(K)

such that

(
h
(
α
)
,
p
)
=
1

(h(\alpha ),p) = 1

. As immediate corollaries of this we get a result of Toda on the homotopy groups of the reduced product spaces of spheres and a well-known result of Serre on the odd primary parts of the homotopy groups of spheres.
On Kneser’s addition theorem in groups
Authors: Diderrich George T. 
Published: June 2010

The following theorem is proved. Theorem A. Let

G
G

be a group written additively with finite nonempty subsets

A
,
B

A,B

. Assume that

B
B

is commutative, i.e.

b
1

+

b
2

=

b
2

+

b
1

{b_1} + {b_2} = {b_2} + {b_1}

, for

b
1

,

b
2

∈
B

{b_1},{b_2} \in B

. Then there exists an Abelian subgroup

H
H

of

G
G

such that

A
+
B
+
H
=
A
+
H
+
B
=
A
+
B

A + B + H = A + H + B = A + B

and

|

A
+
B

|

≧

|

A
+
H

|

+

|

B
+
H

|

−

|

H

|

|A + B| \geqq |A + H| + |B + H| - |H|

. This is Kneser’s theorem, if

G
G

is Abelian. Also, as an application of the above theorem, the following is proved. Theorem B. Let

G
G

be a finite group of order

υ
(
υ
>
1
)

\upsilon (\upsilon > 1)

and let

a
1

,
⋯
,

a
υ

{a_1}, \cdots ,{a_\upsilon }

be a sequence (repeats are allowed) of nonzero elements of

G
G

. The set

S
S

of sums

a

i
1

+
⋯
+

a

i
t

{a_{{i_1}}} + \cdots + {a_{{i_t}}}

where

1
≦

i
1

>
⋯
>

i
t

≦
υ

1 \leqq {i_1} > \cdots > {i_t} \leqq \upsilon

and

1
≦
t
≦
υ

1 \leqq t \leqq \upsilon

must contain a nontrivial subgroup

H
H

of

G
G

. Finally, the Kemperman

d
d

-transform, a transform similar to the Dyson

e
e

-transform, is introduced and evidence is given to support the conjecture that Theorem A remains true, if the commutativity of

B
B

is dropped.
Lower bounds for solutions of hyperbolic inequalities in unbounded regions
Authors: Murray Amy C. 
Published: June 2010

This paper considers

C
2

{C^2}

solutions

u
=
u
(
t
,
x
)

u = u(t,x)

of the differential inequality

|

L
u

|

≦

k
1

(
t
,
x
)

|

u

|

+

k
2

(
t
,
x
)

|

|

∇
u

|

|

|Lu| \leqq {k_1}(t,x)|u| + {k_2}(t,x)||\nabla u||

. The coefficients of the hyperbolic operator

L
L

depend on both

t
t

and

x
x

. Explicit lower bounds are given for the energy of

u
u

in a region of

x
x

-space expanding at least as fast as wave-fronts for

L
L

. These bounds depend on the asymptotic behavior of

k
1

,

k
2

{k_1},{k_2}

, and the coefficients of

L
L

. They do not require boundary conditions on

u
u

.
Ideals of square summable power series in several variables
Authors: Radlow James 
Published: June 2010

Let

C

(
z
)

\mathcal {C}(z)

be the Hilbert space of formal power series in

z
1

,
⋯
,

z
r

(
r
≧
1
)

{z_1}, \cdots ,{z_r}(r \geqq 1)

. An ideal of

C

(
z
)

\mathcal {C}(z)

is a vector subspace

M

\mathcal {M}

of

C

(
z
)

\mathcal {C}(z)

which contains

z
1

f
(
z
)
,
⋯
,

z
r

f
(
z
)

{z_1}f(z), \cdots ,{z_r}f(z)

whenever it contains

f
(
z
)

f(z)

. If

B
(
z
)

B(z)

is a formal power series such that

B
(
z
)
f
(
z
)

B(z)f(z)

belongs to

C

(
z
)

\mathcal {C}(z)

and

|

|

B
(
z
)
f
(
z
)

|

|

=

|

|

f
(
z
)

|

|

||B(z)f(z)|| = ||f(z)||

, then the set

M

(
B
)

\mathcal {M}(B)

of all products

B
(
z
)
f
(
z
)

B(z)f(z)

is a closed ideal of

C

(
z
)

\mathcal {C}(z)

. In the case

r
=
1

r = 1

, Beurling showed that every closed ideal is of this form for some such

B
(
z
)

B(z)

. Here we give conditions under which a closed ideal is of the form

M

(
B
)

\mathcal {M}(B)

for

r
≧
2

r \geqq 2

.
On normal derivations
Authors: Anderson Joel 
Published: June 2010

Let

Δ
T

{\Delta _T}

be the derivation on

B

(

H

)

\mathfrak {B}(\mathcal {H})

defined by

Δ
T

(
X
)
=
T
X
−
X
T
(
T
,
X
∈

B

(

H

)
)

{\Delta _T}(X) = TX - XT(T,X \in \mathfrak {B}(\mathcal {H}))

. We prove that if

T
T

is an isometry or a normal operator, then the range of

Δ
T

{\Delta _T}

is orthogonal to the null space of

Δ
T

{\Delta _T}

. Also, we prove that if

T
T

is normal with an infinite number of points in its spectrum then the closed linear span of the range and the null space of

Δ
T

{\Delta _T}

is not all of

B

(

H

)

\mathfrak {B}(\mathcal {H})

.
Bounded, conservative, linear operators and the maximal group. II
Authors: Kelly E. P. , Hogan D. A. 
Published: June 2010

Let

V
V

denote an infinite dimensional Banach space over the complex field,

B
[
V
]

B[V]

the bounded linear operators on

V
V

and

F
F

a closed subspace of

V
V

. An element of

T

F

=
{
T

|

T
∈
B
[
V
]
,
T
(
F
)
⊆
F
}

{\mathcal {T}_F} = \{ T|T \in B[V],T(F) \subseteq F\}

is called a conservative operator. Some sufficient conditions for

T
∈

T

F

T \in {\mathcal {T}_F}

to be in the boundary,

B

\mathcal {B}

, of the maximal group,

M

\mathcal {M}

, of invertible elements are determined. For example, if

T
∈

T

F

T \in {\mathcal {T}_F}

, is such that (i)

V
V

is the topological direct sum of

R

(
T
)

\mathcal {R}(T)

and

N
(
T
)
≠
{
θ
}

N(T) \ne \{ \theta \}

, (ii)

T
T

is an automorphism on

R

(
T
)
∩
F

\mathcal {R}(T) \cap F

, then

T
∈

B

T \in \mathcal {B}

. Also, the complement of the closure of

M

\mathcal {M}

is discussed. This is an extension of another paper by the same authors [6].
A two-color theorem for analytic maps in 𝑅ⁿ
Authors: Cain Bryan E. 
Published: June 2010

f
1

,
⋯
,

f
k

{f_1}, \cdots ,{f_k}

are real analytic functions on

R
n

{R^n}

th then the connected components of

R
n

∖
[

f
1

−
1

(
0
)
∪
⋯
∪

f
k

−
1

(
0
)
]

{R^n}\backslash [f_1^{ - 1}(0) \cup \cdots \cup f_k^{ - 1}(0)]

can be “colored” with two colors so that two components will have different colors whenever their common boundary contains a topological

(
n
−
1
)

(n - 1)

-manifold.
Asymptotic inversion of Laplace transforms: a class of counterexamples
Authors: Lew John S. 
Published: June 2010

Let

f
f

be a complex-valued locally integrable function on

[
0
,
+
∞
)

[0, + \infty )

, and let

L
f

Lf

be its Laplace transform, whenever and wherever it exists. We review some known methods, exact and approximate, for recovering

f
f

from

L
f

Lf

. Since numerical algorithms need auxiliary information about

f
f

near

+
∞

+ \infty

, we note that the behavior of

f
f

near

+
∞

+ \infty

depends on the behavior of

L
f

Lf

near 0 +, hence that our ability to retrieve

f
f

is limited by the class of momentless functions, namely, all functions

f
f

such that

L
f
(
s
)

Lf(s)

converges absolutely for

Re
⁡
(
s
)
>
0

\operatorname {Re} (s) > 0

and satisfies
\[

L
f
(
s
)
=
o
(

s
n

)

near

0
+

for

n
=
0
,
1
,
2
,
⋯
.

Lf(s) = o({s^n}){\text { near }}0 + \quad {\text {for}}\;n = 0,1,2, \cdots .

\]
We discuss the space

Z
Z

of momentless functions and complex distributions, then construct a family of elements in this space which defy various plausible conjectures.
The fundamental ideal and 𝜋₂ of higher dimensional knots
Authors: Lomonaco S. J. 
Published: June 2010

Let

(

S
4

,
k
(

S
2

)
)

({S^4},k({S^2}))

be a knot formed by spinning a polyhedral arc

α
\alpha

about the standard

2
2

-sphere

S
2

{S^2}

in the

3
3

-sphere

S
3

{S^3}

. Then the second homotopy group of

S
4

−
k
(

S
2

)

{S^4} - k({S^2})

as a

Z

π
1

Z{\pi _1}

-module is isomorphic to each of the following: (1) The fundamental ideal modulo the left ideal generated by

a
−
1

a - 1

, where

a
a

is the image in

π
1

(

S
4

−
k
(

S
2

)
)

{\pi _1}({S^4} - k({S^2}))

of a generator of

π
1

(

S
2

−
α
)

{\pi _1}({S^2} - \alpha )

. (2) The first homology group of the kernel of

π
1

(

S
3

−
k
(

S
2

)
)
→

π
1

(

S
4

−
k
(

S
2

)
)

{\pi _1}({S^3} - k({S^2})) \to {\pi _1}({S^4} - k({S^2}))
Criteria for compactness and for discreteness of locally compact amenable groups
Authors: Granirer Edmond 
Published: June 2010

Let

G
G

be a locally compact group

P
(
G
)
=
{
0
≦
ϕ
∈

L
1

(
G
)
;
∫

ϕ
(
x
)
d
x
=
1
}

P(G) = \{ 0 \leqq \phi \in {L_1}(G);\int {\phi (x)dx = 1\} }

and

(

l
a

f
)
(
x
)
=

a

f
(
x
)
=
f
(
a
x
)

({l_a}f)(x) = {}_af(x) = f(ax)

for all

a
,
x
∈
G

a,x \in G

and

f
∈

L
∞

(
G
)
.0
≦
Ψ
∈

L
∞

(
G

)
∗

,
Ψ
(
1
)
=
1

f \in {L^\infty }(G).0 \leqq \Psi \in {L^\infty }{(G)^ \ast },\Psi (1) = 1

is said to be a [topological] left invariant mean ([TLIM] LIM) if

Ψ

(

a

f
)
=
Ψ
(
f
)
[
Ψ
(
ϕ
∗
f
)
=
Ψ
(
f
)

\Psi {{\text {(}}_a}f) = \Psi (f)[\Psi (\phi \ast f) = \Psi (f)

] for all

a
∈
G
,
ϕ
∈
P
(
G
)
,
f
∈

L
∞

(
G
)

a \in G,\phi \in P(G),f \in {L^\infty }(G)

. The main result of this paper is the Theorem. Let

G
G

be a locally compact group, amenable as a discrete group. If

G
G

contains an open

σ
\sigma

-compact normal subgroup, then LIM = TLIM if and only if

G
G

is discrete. In particular if

G
G

is an infinite compact amenable as discrete group then there exists some

Ψ
∈
L
I
M

\Psi \in LIM

which is different from normalized Haar measure. A harmonic analysis type interpretation of this and related results are given at the end of this paper.

2

^{2}
𝑛-normal lattices
Authors: Cornish William H. 
Published: June 2010

An

n
n

-normal lattice is a distributive lattice with 0 such that each prime ideal contains at most

n
n

minimal prime ideals. A relatively

n
n

-normal lattice is a distributive lattice such that each bounded closed interval is an

n
n

-normal lattice. The main results of this paper are: (1) a distributive lattice

L
L

with 0 is

n
n

-normal if and only if for any

x
0

,

x
1

,
⋯
,

x
n

ε
L

{x_0},{x_1}, \cdots ,{x_n}\varepsilon L

such that

x
i

∧

x
j

=
0

{x_i} \wedge {x_j} = 0

for any

i
≠
j
,
i
,
j
=
0
,
⋯
,
n
,

(

x
0

]
∗

∨

(

x
1

]
∗

∨
⋯
∨

(

x
n

]
∗

=
L

i \ne j,i,j = 0, \cdots ,n,{({x_0}]^ \ast } \vee {({x_1}]^ \ast } \vee \cdots \vee {({x_n}]^ \ast } = L

, (2) a distributive lattice

L
L

is relatively

n
n

-normal if and only if for any

n
+
1

n + 1

incomparable prime ideals

P
0

,

P
1

,
…
,

P
n

,

P
0

∨

P
1

∨
…
∨

P
n

=
L

{P_0},{P_1}, \ldots ,{P_n},{P_0} \vee {P_1} \vee \ldots \vee {P_n} = L

.
Eilenberg-Mac Lane spectra
Authors: Margolis H. R. 
Published: June 2010

Let

K
(

Z
p

)

K({Z_p})

be the Eilenberg-Mac Lane spectrum with homotopy

Z
p

{Z_p}

and let

A
=

H
∗

(
K
(

Z
p

)
;

Z
p

)

A = {H^ \ast }(K({Z_p});{Z_p})

—the

mod
p

\bmod p

Steenrod algebra. Let

X
X

be a locally finite spectrum. It is proven that
\[

[
K
(

Z
p

)
,
X
]
→

Hom
A

(

H
∗

(
X
;

Z
p

)
,
A
)

[K({Z_p}),X] \to {\operatorname {Hom} _A}({H^ \ast }(X;{Z_p}),A)

\]
is an isomorphism. It is also proven that there is a unique decomposition

X
=
(
⊕
K
(

Z
p

)
)
⊕
Y

X = ( \oplus K({Z_p})) \oplus Y

where

H
∗

(
Y
;

Z
p

)

{H^ \ast }(Y;{Z_p})

as an

A
A

-module has no free summands.
The Riesz summability of logarithmic type
Authors: Kwee B. 
Published: June 2010

The series

Σ

n
=
1

∞

a
n

\Sigma _{n = 1}^\infty {a_n}

is said to be summable

(
L
)

(L)

to

s
s

if

(
log
⁡
(
1
−
x
)

)

−
1

Σ

n
=
1

∞

s
n

x

n
+
1

/

n

{(\log (1 - x))^{ - 1}}\Sigma _{n = 1}^\infty {s_n}{x^{n + 1}}/n

, where

s
n

=

Σ

v
=
1

n

a
v

{s_n} = \Sigma _{v = 1}^n{a_v}

, converges for

0
≤
x
>
1

0 \leq x > 1

and tends to

s
s

when

x
→
1
−

x \to 1 -

. The aim of this paper is to discuss the relation between summability

(
L
)

(L)

and Riesz summability

(
R
,
log
⁡
n
,
κ
)

(R,\log n,\kappa )

. It is proved that

(
R
,
log
⁡
n
,
κ
)
⊆
(
L
)

(R,\log n,\kappa ) \subseteq (L)

holds for

0
≤
κ
≤
1

0 \leq \kappa \leq 1

and is false for

κ
>
1

\kappa > 1

. It is also proved that if

Σ

n
=
1

∞

a
n

=
s
(
L
)

\Sigma _{n = 1}^\infty {a_n} = s(L)

and bounded

(
R
,
log
⁡
n
,
κ
)

(R,\log n,\kappa )

for

κ
≥
0

\kappa \geq 0

then

Σ

n
=
1

∞

a
n

=
s
(
R
,
log
⁡
n
,
κ
+
δ
)

\Sigma _{n = 1}^\infty {a_n} = s(R,\log n,\kappa + \delta )

for every

δ
>
0

\delta > 0

.
A comparison theorem for certain functional differential equations
Authors: True Ernest D. 
Published: June 2010

The oscillatory character of solutions to the functional differential equation

x

(
n
)

(
t
)
+
a
(
t
)
f
(
x
(
g
(
t
)
)
)
=
Q
(
t
)

{x^{(n)}}(t) + a(t)f(x(g(t))) = Q(t)

is investigated, by comparison with the oscillatory character of solutions to

x

(
n
)

(
t
)
+
s
(
t
)
f
(
x
(
t
)
)
=
0

{x^{(n)}}(t) + s(t)f(x(t)) = 0

where

s
(
t
)
⩾
γ
a
(
t
)
,
0
>
γ
>
1

s(t) \geqslant \gamma a(t),0 > \gamma > 1

. Here,

Q
(
t
)

Q(t)

represents a bounded, oscillatory forcing function, and

g
(
t
)

g(t)

tends to

∞
\infty

as

t
→
∞

t \to \infty

or

g
(
t
)
⩾
t
−
c

g(t) \geqslant t - c

for large

t
t

but is otherwise arbitrary.
Whitney number inequalities for geometric lattices
Authors: Dowling Thomas A. , Wilson Richard M. 
Published: June 2010

Let

L
L

be a finite geometric lattice of rank

r
r

, and for

i
=
0
,
1
,
⋯
,
r

i = 0,1, \cdots ,r

, let

W
i

{W_i}

denote the number of elements of

L
L

with rank

i
i

. For

1
≤
k
≤
r
−
2

1 \leq k \leq r - 2

, we have

W
1

+

W
2

+
⋯
+

W
k

≤

W

r
−
k

+
⋯
+

W

r
−
2

+

W

r
−
1

{W_1} + {W_2} + \cdots + {W_k} \leq {W_{r - k}} + \cdots + {W_{r - 2}} + {W_{r - 1}}

with equality if and only if the lattice

L
L

is modular. We give two further results concerning matchings of lattice elements of rank

≤
k

\leq k

into those of rank

≥
r
−
k

\geq r - k

, and observe that a middle term can be interpolated in the above inequality.
𝜙-Postnikov systems and extensions of 𝐻-spaces
Authors: Shar Albert O. 
Published: June 2010

Let

f
:
X
→
Y

f:X \to Y

be a map of CW complexes and let

π
:

F
f

→
X

\pi :{F_f} \to X

be the fibration induced by

f
f

. The following theorems are proven: Theorem. Assume

F
(
=

F
f

)

F( = {F_f})

and

Ω
Y

\Omega Y

are simply connected and that (a)

f
∗

:

H
n

(
Y
;

π
n

(
Y
)
)
→

H
n

(
X
;

π
n

(
Y
)
)

{f^ \ast }:{H^n}(Y;{\pi _n}(Y)) \to {H^n}(X;{\pi _n}(Y))

is epic for all

n
n

, (b)

(
i
∧
i

)
∗

:

H
n

(
F
∧
F
;
co
⁡
ker
⁡

f

n
∗

)
→

H
n

(
Ω
Y
∧
Ω
Y
;
co
⁡
ker
⁡

f

n
∗

)

{(i \wedge i)^ \ast }:{H^n}(F \wedge F;\operatorname {co} \ker {f_{{n^ \ast }}}) \to {H^n}(\Omega Y \wedge \Omega Y;\operatorname {co} \ker {f_{{n^ \ast }}})

is monic for all

n
n

(where

f

n
∗

:

π
n

(
X
)
→

π
n

(
Y
)
)

{f_{{n^ \ast }}}:{\pi _n}(X) \to {\pi _n}(Y))

). If

X
X

is an

H
H

-space then

F
F

is an

H
H

-space such that

π
:
F
→
X

\pi :F \to X

is an

H
H

-map. Theorem. Assume

Y
Y

is

(
p
−
1
)

(p - 1)

-connected,

F
F

is

(
q
−
1
)

(q - 1)

-connected

(
p
−
1
≥
2
,
q
−
1
≥
1
)

(p - 1 \geq 2,q - 1 \geq 1)

of dimension

>
min
(
2
p
−
1
,
p
+
q
−
1
)

> \min (2p - 1,p + q - 1)

, and

H
∗

(
Y
)

{H^ \ast }(Y)

is free. If

X
X

is an

H
H

-space and

f
∗

:

H
n

(
Y
)
→

H
n

(
X
)

{f^ \ast }:{H^n}(Y) \to {H^n}(X)

is onto for all

n
n

then

F
F

is an

H
H

-space and the map

π
:
F
→
X

\pi :F \to X

is an

H
H

-map. Analogous theorems are shown to hold for loop spaces.
Asymptotic behavior and nonoscillation of Volterra integral equations and functional differential equations
Authors: Izé A. F. , Freiria A. A. 
Published: June 2010

It is proved that if

q

i
j

(
t
,
s
)

ρ
j

(
s
)

[

ρ
i

(
t
)

]

−
1

{q_{ij}}(t,s){\rho _j}(s){[{\rho _i}(t)]^{ - 1}}

is bounded,

i
,
j
=
1
,
2
,
…
,
n

i,j = 1,2, \ldots ,n

, and

f
(
t
,
x
,
x
(
u
(
s
)
)
)

f(t,x,x(u(s)))

is “small",
\[

x
(
u
(
s
)
)
=
(

x
1

(

u
1

(
s
)
)
,

x
2

(

u
2

(
s
)
)
,
…
,

x
n

(

u
n

(
s
)
)
)

x(u(s)) = ({x_1}({u_1}(s)),{x_2}({u_2}(s)), \ldots ,{x_n}({u_n}(s)))

\]
with

u
i

(
t
)
⩽
t

{u_i}(t) \leqslant t

and

lim

t
→
∞

u
i

(
t
)
=
∞

{\lim _{t \to \infty }}{u_i}(t) = \infty

, the solutions of the integral equation
\[

x

(
t
)

=
h
(
t
)
+

∫
0
t

q
(
t
,
s
)
f
(
s
,
x
(
s
)
,
x
(
u
(
s
)
)
)
d
s

x\left ( t \right ) = h(t) + \int _0^t {q(t,s)f(s,x(s),x(u(s)))ds}

\]
satisfy the conditions

x
(
t
)
=
h
(
t
)
+
ρ
(
t
)
a
(
t
)
,

lim

t
→
∞

a
(
t
)
=

x(t) = h(t) + \rho (t)a(t),{\lim _{t \to \infty }}a(t) =

constant where

ρ
(
t
)

\rho (t)

is a nonsingular diagonal matrix chosen in such a way that

ρ

−
1

(
t
)
h
(
t
)

{\rho ^{ - 1}}(t)h(t)

is bounded. The results contain, in particular, some results on the asymptotic behavior, stability and existence of nonoscillatory solutions of functional differential equations.
Representation of a crinkled arc
Authors: Vitale Richard A. 
Published: June 2010

Johnson [A crinkled arc, Proc. Amer. Math. Soc. 25 (1970), 375-376] has shown that under suitable normalizations all crinkled arcs are unitarily equivalent. Using this result, we find a general series expansion for a crinkled arc:
\[

f
(
t
)
=

2

∑

n
=
1

∞

x
n

sin
⁡
(
n
−
1

/

2
)
π
t

(
n
−
1

/

2
)
π

,

f(t) = \sqrt {2} \sum _{n=1}^\infty x_n \tfrac {\sin (n - 1/2)\pi t}{(n - 1/2)\pi } ,

\]
where

{

x
n

}

\{ {x_n}\}

is an orthonormal set.
A geometric property of certain plane sets
Authors: Pietz Kenneth 
Published: June 2010

Suppose

K
K

is a compact subset of the plane of the form

Δ
(
0
,
1
)

¯

∖

∪

n
=
1

∞

Δ
(

p
n

,

r
n

)

\overline {\Delta (0,1)} \backslash \cup _{n = 1}^\infty \Delta ({p_n},{r_n})

where

Δ
(

p
n

,

r
n

)

¯

⊆
Δ
(
0
,
1
)

\overline {\Delta ({p_n},{r_n})} \subseteq \Delta (0,1)

for each

n
n

and

Δ
(

p
i

,

r
i

)

¯

∩

Δ
(

p
j

,

r
j

)

¯

=
∅

\overline {\Delta ({p_i},{r_i})} \cap \overline {\Delta ({p_j},{r_j})} = \emptyset

for

i
≠
j

i \ne j

. Let

α
=

sup

i
⩾
1

(
(

r
i

+
1
)

/

r
i

)

\alpha = {\sup _{i \geqslant 1}}(({r_i} + 1)/{r_i})

and define the sets

∂

∗

K
≡
∂
Δ
(
0
,
1
)
∪
[

∪

n
=
1

∞

∂
Δ
(

p
n

,

r
n

)
]

{\partial _{\ast }}K \equiv \partial \Delta (0,1) \cup [ \cup _{n = 1}^\infty \partial \Delta ({p_n},{r_n})]

and

F
(
K
)
≡
{
z
∈
K
∖

∂

∗

K
=
z

is not a point of density of

K
}

F(K) \equiv \{ z \in K\backslash {\partial _{\ast }}K = z{\text { is not a point of density of }}K\}

. It is proved that if

α
>
1

\alpha > 1

, then

K

1

[
F
(
K
)
]
=
0

{\mathcal {K}^1}[F(K)] = 0

, where

K

1

{\mathcal {K}^1}

denotes Hausdorff one-dimensional measure.
On a conjecture of S. Chowla
Authors: Suryanarayana D. 
Published: June 2010

Let

ψ
(
x
)
=
x
−
[
x
]
−

1
2

\psi (x) = x - [x] - \tfrac {1}{2}

. It has been conjectured by S. Chowla that

Σ

n
⩽
√
x

{

ψ
2

(
x

/

n
)
−
1

/

12
}
=
o
(

x

1

/

4
+
ϵ

)

{\Sigma _{n \leqslant \surd x}}\{ {\psi ^2}(x/n) - 1/12\} = o({x^{1/4 + \epsilon }})

, for every

ϵ
>
0

\epsilon > 0

. In this paper we show that this conjecture is equivalent to

Σ

n
⩽
√
x

n
2

{

ψ
2

(
x

/

n
)
−
1

/

12
}

{\Sigma _{n \leqslant \surd x}}{n^2}\{ {\psi ^2}(x/n) - 1/12\}

by proving that
\[

∑

n
⩽
√
x

{

ψ
2

(

x
n

)

−

1

12

}

+

1
x

∑

n
⩽
√
x

n
2

{

ψ
2

(

x
n

)

−

1

12

}

=
o
(

x

1

/

4

)
.

\sum \limits _{n \leqslant \surd x} {\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} + \tfrac {1}{x}\sum \limits _{n \leqslant \surd x} {{n^2}\left \{ {{\psi ^2}\left ( {\tfrac {x}{n}} \right ) - \tfrac {1}{{12}}} \right \} = o({x^{1/4}}).} }

\]