bluuebunny/crossref_metadata_2025_split huggingface dataset:
Imbeddings, immersions, and characteristic classes of differentiable manifolds
Authors: Papastavridis Stavros 
Published: June 2010

Let

I
n
i

I_n^i

be the set of

mod

-

2

\bmod {\text { - }}2

characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let

I

n
,
k

i

I_{n,k}^i

be the set of i-dimensional

mod

-

2

\bmod {\text { - }}2

characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in

H
∗

(
B
O
,

Z
2

)

{H^ \ast }(BO,{Z_2})

generated by

w

k
+
1

,

w

k
+
2

,
…

{w_{k + 1}},{w_{k + 2}}, \ldots

. Then if

i
⩽
(
n
+
k
)

/

2

i \leqslant (n + k)/2

, we have

I

n
,
k

i

=

I
n
i

+

K
i

I_{n,k}^i = I_n^i + {K^i}

. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.
Vector-valued continuous functions with strict topologies and angelic topological spaces
Authors: Khurana Surjit Singh 
Published: June 2010

It is proved that if X is a metric space, E a Banach space containing a

σ
\sigma

-weakly-compact dense subset, then the space

(

M
τ

(
X
,

E
′

)
,
σ
(

M
τ

(
X
,

E
′

)
,

C
b

(
X
,
E
)
)
)

({M_\tau }(X,E’),\sigma ({M_\tau }(X,E’),{C_b}(X,E)))

is angelic,

C
b

(
X
,
E
)

{C_b}(X,E)

being all bounded continuous functions from X into E and

M
τ

(
X
,

E
′

)

{M_\tau }(X,E’)

the dual of

C
b

(
X
,
E
)

{C_b}(X,E)

with the strict topology

β
\beta

.
The Cauchy problem for hyperbolic operators with variable multiple characteristics
Authors: Yamamoto Kazuhiro 
Published: June 2010

Let

P
(
t
,
x
,

D
t

,

D
x

)

P(t,x,{D_t},{D_x})

be a hyperbolic differential operator with the principal symbol

p
m

(
t
,
x
,
τ
,
ξ
)

{p_m}(t,x,\tau ,\xi )

. We assume that

P
m

{P_m}

is denoted by

Π

j
=
1

s

(
τ
−

λ
j

)

m
j

Π

j
=
s
+
1

m
−
N
+
s

(
τ
−

λ
j

)

\Pi _{j = 1}^s{(\tau - {\lambda _j})^{{m_j}}}\Pi _{j = s + 1}^{m - N + s}(\tau - {\lambda _j})

and

(

λ
i

−

λ
j

)
(
t
,
x
,
ξ
)
≠
0

({\lambda _i} - {\lambda _j})(t,x,\xi ) \ne 0

if

(
i
,
j
)
≠
(
k
,
m
−
N
+
k
)

(
k
=
1
,
…
,
s
)

(i,j) \ne (k,m - N + k)\;(k = 1, \ldots ,s)

, where

N
=

Σ

j
=
1

s

m
j

N = \Sigma _{j = 1}^s{m_j}

and

λ
j

(
t
,
x
,
ξ
)
∈

C
∞

(
[
0
,
T
]
×

R
n

×
(

R
n

∖
0
)
)

{\lambda _j}(t,x,\xi ) \in {C^\infty }([0,T] \times {R^n} \times ({R^n}\backslash 0))

. Under a generalized condition of E. E. Levi, we shall show that the Cauchy problem

P
u
=
f

Pu = f

in

[
0
,
T
]
×

R
n

,

D
t
j

u

|

t
=
0

=

g
j

(
j
=
1
,
…
,
m
−
1
)

[0,T] \times {R^n},D_t^j{u_{|t = 0}} = {g_j}(j = 1, \ldots ,m - 1)

is well posed. When

m
j

=
1
(
j
=
1
,
…
,
s
)

{m_j} = 1(j = 1, \ldots ,s)

, our result coincides those of Ohya and Petkov.
Capacities and spans on Riemann surfaces
Authors: Burbea Jacob 
Published: June 2010

Let

K
(
z
,
z
)
,
R
(
z
,
z
)

K(z,z),R(z,z)

, and

C
(
z
)

C(z)

be the values of the Bergman kernel, the reduced Bergman kernel and the analytic capacity on an open Riemann surface

Ω
\Omega

(with respect to a local parameter z). Let

M
(
z
)
=
π
K
(
z
,
z
)

M(z) = \pi K(z,z)

and

S
(
z
)
=

π
R
(
z
,
z
)

S(z) = \sqrt {\pi R(z,z)}

. For

Ω
∉

O
G

\Omega \notin {O_G}

and for each integer

n
⩾
0

n \geqslant 0

, it is shown that
\[

C

(
n
+
1
)
(
n
+
2
)

⩽
(
n
+
1
)
!

(

∏

k
=
0

n
+
1

k
!

)

−
2

det

‖

M

j

k
¯

‖

j
,
k
=
0

n

,

{C^{(n + 1)(n + 2)}} \leqslant (n + 1)!{\left ( {\prod \limits _{k = 0}^{n + 1} {k!} } \right )^{ - 2}}\det \left \| {{M_{j\bar k}}} \right \|_{j,k = 0}^n,

\]
where

C
=
C
(
z
)

C = C(z)

and

M

j

k
¯

=
(

∂

j
+
k

/

∂

z
j

∂

z
¯

k

)
M
(
z
)

{M_{j\bar k}} = ({\partial ^{j + k}}/\partial {z^j}\partial {\bar z^k})M(z)

. Equality occurs if and only if

Ω
\Omega

is conformally equivalent to the unit disk less (possibly) a closed set of inner capacity zero. The special case of this result, namely when

n
=
0

n = 0

, is due to Hejhal and Suita. Let

κ
(
z
)

\kappa (z)

be the curvature of the “span metric”

S
(
z
)

|

d
z

|

S(z)|dz|

. As an attempt to resolve a conjecture of Suita, we also show that for

Ω
∉

O

A
D

,
κ
(
z
)
⩽
−
2

\Omega \notin {O_{AD}},\kappa (z) \leqslant - 2

for each

z
∈
Ω

z \in \Omega

. Both results are proved by studying suitable extremal problems.
Differentiable pseudo-free circle actions on homotopy spheres
Authors: Liang Chao Chu 
Published: June 2010

Let G denote the circle group,

φ
\varphi

a differentiable pseudo-free G-action of type

(

p
1

,
…
,

p
k

)

({p_1}, \ldots ,{p_k})

on a homotopy sphere

Σ

2
n
+
1

{\Sigma ^{2n + 1}}

, and X the vector field induced by

φ
\varphi

. If w is a G-invariant 1-form satisfying

w
(
X
)
=
1

w(X) = 1

, then we will prove that

∫
Σ

w
∧

(
d
w

)
n

=
±

(

p
1

p
2

⋯

p
k

)

−
1

{\smallint _\Sigma }w \wedge {(dw)^n} = \pm {({p_1}{p_2} \cdots {p_k})^{ - 1}}

.
Two applications of change of rings theorems for Poincaré series
Authors: Herzog Jürgen , Steurich Manfred 
Published: June 2010

Let (R, m, k) be an artinian Gorenstein ring with

dim
k

m

/

m
2

⩾
2

{\dim _k}m/{m^2} \geqslant 2

. If

P
R

{P^R}

denotes the Poincaré-series and

ϕ
R

{\phi _R}

denotes the Bass-series of R, then

ϕ
R

=
(

P
R

−
1
)

x

−
1

(
1
−

P
R

x
2

)

−
1

{\phi _R} = ({P^R} - 1){x^{ - 1}}{(1 - {P^R}{x^2})^{ - 1}}

with

R
¯

=
R

/

0
:
m

\bar R = R/0:m

, see Proposition 1. With similar methods as in the proof of Proposition 1 we calculate the Poincaré-series of a certain class of local rings: Let (R, m, k) be a local ring,

(

α

i
j

)

({\alpha _{ij}})

a

n
−
1
×
n

n - 1 \times n

-matrix with entries in m,

n
⩾
3

n \geqslant 3

. Let

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

be elements in m and

A

\mathfrak {A}

the ideal generated by the maximal minors of

(

α

i
j

)

({\alpha _{ij}})

and by

Σ
j

α

i
j

x
j

(
i
=
1
,
…
,
n
−
1
)

{\Sigma _j}{\alpha _{ij}}{x_j}(i = 1, \ldots ,n - 1)

. If depth

A

=
n

\mathfrak {A} = n

, then

P

R

/

A

⋅
(

(
1
−
X

)

n
−
1

−
X
)

(
1
+
X

)
n

=

P
R

{P^{R/\mathfrak {A}}} \cdot ({(1 - X)^{n - 1}} - X){(1 + X)^n} = {P^R}

.
A Helly-type theorem for the dimension of the kernel of a starshaped set
Authors: Breen Marilyn 
Published: June 2010

This study will investigate the dimension of the kernel of a starshaped set, and the following result will be obtained: Let S be a compact set in some linear topological space L. For

1
⩽
k
⩽
n

1 \leqslant k \leqslant n

, the dimension of

ker
⁡
S

\ker S

is at least k if and only if for some

ε
>
0

\varepsilon > 0

and some n-dimensional flat

F
n

{F^n}

in L, every

f
(
n
,
k
)

f(n,k)

points of S see via S a common k-dimensional neighborhood in

F
n

{F^n}

having radius

ε
\varepsilon

. The number

f
(
n
,
k
)

f(n,k)

is defined inductively as follows:
\[

f
(
2
,
1
)
=
4
,

f
(
n
,
k
)
=
f
(
n
−
1
,
k
)
+
n
+
2

for

3
⩽
n

and

1
⩽
k
⩽
n
−
1
,

f
(
n
,
n
)
=
n
+
1.

\begin {array}{*{20}{c}} {f(2,1) = 4,} \hfill \\ {f(n,k) = f(n - 1,k) + n + 2\quad {\text {for}}\;3 \leqslant n\;{\text {and}}\;1 \leqslant k \leqslant n - 1,} \hfill \\ {f(n,n) = n + 1.} \hfill \\ \end {array}

\]
An integral representation of holomorphic functions
Authors: Hilger R. W. , Michaliček J. F. 
Published: June 2010

Let K be a compact set in the complex plane and let f be a function holomorphic on the complement

Ω
\Omega

of K and vanishing at infinity. We prove that there are finite complex-valued Borel measures

μ

m
,
n

(
m
,
n
=
0
,
1
,
2
,
…
;
m
+
n
⩾
1
)

{\mu _{m,n}}(m,n = 0,1,2, \ldots ;m + n \geqslant 1)

on

K
2

{K^2}

satisfying

lim

k
→
∞

(

Σ

m
+
n
=
k

‖

μ

m
,
n

‖

)

1

/

k

=
0

{\lim _{k \to \infty }}{({\Sigma _{m + n = k}}\left \|{\mu _{m,n}}\right \|)^{1/k}} = 0

so that
\[

f
(
z
)
=

∑

m
,
n

∫

K
2

(
z
−

w
1

)

−
m

(
z
−

w
2

)

−
n

d

μ

m
,
n

(

w
1

,

w
2

)

(
z
∈
Ω
)
.

f(z) = \sum \limits _{m,n} {\int _{{K^2}} {{{(z - {w_1})}^{ - m}}{{(z - {w_2})}^{ - n}}d{\mu _{m,n}}({w_1},{w_2})\quad (z \in \Omega ).} }

\]
Convergence of measurable random functions
Authors: Grinblat L. Š. 
Published: June 2010

Using the theorem of Fréchet and Kolmogorov about compact subsets of the space

L
p

[
0
,
1
]

{L_p}[0,1]

and Prohorov’s theorem about the convergence of measures defined on a complete metric space we proved in [2] the following theorem: Let

ξ
1

(
t
)
,
…
,

ξ
n

(
t
)
,
…

{\xi _1}(t), \ldots ,{\xi _n}(t), \ldots

and

ξ
(
t
)

\xi (t)

be measurable random processes

(
0
⩽
t
⩽
1
)

(0 \leqslant t \leqslant 1)

and suppose that there exist numbers C and

p
⩾
1

p \geqslant 1

such that

E

|

ξ
n

(
t
)

|

p

⩽
C

E|{\xi _n}(t){|^p} \leqslant C

for all n and t. If

E

|

ξ
n

(
t
)

|

p

→
E

|

ξ
(
t
)

|

p

E|{\xi _n}(t){|^p} \to E|\xi (t){|^p}

for all t and if for any finite set

{

t
1

,
…
,

t
k

}
⊂
[
0
,
1
]

\{ {t_1}, \ldots ,{t_k}\} \subset [0,1]

the joint distribution of

ξ
n

(

t
1

)
,
…
,

ξ
n

(

t
k

)

{\xi _n}({t_1}), \ldots ,{\xi _n}({t_k})

converges to the joint distribution of

ξ
(

t
1

)
,
…
,
ξ
(

t
k

)

\xi ({t_1}), \ldots ,\xi ({t_k})

, then the distribution of

f
(

ξ
n

)

f({\xi _n})

converges to the distribution of

f
(
ξ
)

f(\xi )

for any continuous functional f on

L
p

[
0
,
1
]

{L_p}[0,1]

. In this paper this theorem is generalized to random measurable functions. The results of the present paper are related to the results of [1], [3], [4].
The Jacobson radical of the group algebra of a finite group
Authors: Bedi Surinder Singh 
Published: June 2010

Let K be a field of characteristic

p
≠
0

p \ne 0

and G a finite group such that

p

|

o
(
G
)

p|o(G)

. Suppose G is a Frobenius group with a Sylow p-subgroup P as a complement. Then we have proved that
\[

J
K
(
G
)
=

⋂

x
∈
G

J
K
(

P
x

)
K
(
G
)
.

JK(G) = \bigcap \limits _{x \in G} {JK({P^x})K(G).}

\]
We have given an example to show that equality does not hold in general.
Characterization of 𝑝-predictors
Authors: Landers D. , Rogge L. 
Published: June 2010

Let

(
Ω
,

A

,
P
)

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

be a probability space and

1
>
p
>
∞

1 > p > \infty

. It is shown that each operator

T
:

L
p

(
Ω
,

A

,
P
)
→

L
p

(
Ω
,

A

,
P
)

T:{L_p}(\Omega ,\mathcal {A},P) \to {L_p}(\Omega ,\mathcal {A},P)

which is homogeneous, constant preserving, positive, quasi-additive and fulfills Dykstra’s condition is an p-predictor with respect to a suitable

σ
\sigma

-field, i.e. a nearest point projection onto a closed subspace

L
p

(
Ω
,

B

,
P
)

{L_p}(\Omega ,\mathcal {B},P)

, where

B

⊂

A

\mathcal {B} \subset \mathcal {A}

is a

σ
\sigma

-field. None of the conditions for T can be dispensed without compensation.
An Abel-Tauber theorem on convolutions with the Möbius function
Authors: Geluk J. L. 
Published: June 2010

Suppose

n
:

R
+

→

R
+

n:{R^ + } \to {R^ + }

and

n
(
x
)

/

x

n(x)/x

is integrable on

(
0
,
∞
)

(0,\infty )

. For

s
>
0

s > 0

we define
\[

n
~

(
s
)
=
s

∫
0
∞

e

−
u
s

1
−

e

−
u
s

n
(
u
)
d
u
.

\tilde n(s) = s\int _0^\infty {\frac {{{e^{ - us}}}}{{1 - {e^{ - us}}}}} n(u)du.

\]
In this paper an Abel-Tauber theorem is proved concerning this transform. Moreover the relation between

n
~

(
s
)

\tilde n(s)

and

Σ

m
⩽
s

n
(
s

/

m
)

/

m

{\Sigma _{m \leqslant s}}n(s/m)/m

is studied.
The infimum of small subharmonic functions
Authors: Fenton P. C. 
Published: June 2010

Suppose that u is subharmonic in the plane and that, for some

p
>
1
,

lim
_

r
→
∞

B
(
r
)

/

(
log
⁡
r

)
p

=
σ
>
∞

p > 1,{\underline {\lim } _{r \to \infty }}B(r)/{(\log r)^p} = \sigma > \infty

. It is shown that, given

ε
>
0

\varepsilon > 0

,
\[

A
(
r
)
>
B
(
r
)
−
(
σ
+
ε
)
Re
⁡
{

(
log
⁡
r

)
p

−

(
log
⁡
r
+
i
π

)
p

}

A(r) > B(r) - (\sigma + \varepsilon )\operatorname {Re} \{ {(\log r)^p} - {(\log r + i\pi )^p}\}

\]
for r outside an exceptional set E, where
\[

lim
_

x
→
∞

1

(
log
⁡
r
)

p
−
1

∫

E
∩
[
1
,
r
]

(
p
−
1
)

(
log
⁡
t
)

p
−
2

t

d
t

⩽

σ

σ
+
ε

.

\underline {\lim } \limits _{x \to \infty } \;\frac {1}{{{{(\log r)}^{p - 1}}}}\int _{E \cap [1,r]} {\frac {{(p - 1){{(\log t)}^{p - 2}}}}{t}\;dt\; \leqslant \frac {\sigma }{{\sigma + \varepsilon }}.}

\]
An inequality concerning the smallest disc that contains the spectrum of an operator
Authors: Garske Gerhard 
Published: June 2010

Any bounded linear operator T on a Hilbert space satisfies the inequality

sup

‖
x
‖

=
1

{

‖

T
x

‖

2

−

|

(
T
x
,
x
)

|

2

}
⩾

R
T
2

{\sup _{\left \| x \right \| = 1}}\{ {\left \| {Tx} \right \|^2} - |(Tx,x){|^2}\} \geqslant R_T^2

, where

R
T

{R_T}

is the radius of the smallest disc containing the spectrum of T. An example is given, which shows that we cannot expect equality for

G
1

{G_1}

operators in general, contrary to a conjecture of Istrǎţescu.
Topological entropy of block maps
Authors: Coven Ethan M. 
Published: June 2010

We show that

h
(

f
∞

)
=
log
⁡
2

h({f_\infty }) = \log 2

where

f
∞

{f_\infty }

is the map on the space of sequences of zeros and ones induced by the block map

f
(

x
0

,
…
,

x
k

)
=

x
0

+

Π

i
=
1

k

(

x
i

+

b
i

)

f({x_0}, \ldots ,{x_k}) = {x_0} + \Pi _{i = 1}^k({x_i} + {b_i})

where

k
⩾
2

k \geqslant 2

and the k-block

b
1

…

b
k

{b_1} \ldots {b_k}

is aperiodic.
A characterization of normal operators using the Hilbert-Schmidt class
Authors: Nakamoto Ritsuo 
Published: June 2010

A bounded linear operator N on a Hilbert space H is normal if and only if

‖
N
X
−
X
N

‖
2

=
‖

N
∗

X
−
X

N
∗

‖
2

\|NX - XN\|_2 = \|N^* X - X N^*\|_2

for every X in the Hilbert-Schmidt class.
Estimates for exponential sums
Authors: Smith Robert A. 
Published: June 2010

If f is a polynomial over Z of degree

n
+
1

n + 1

with

n
⩾
1

n \geqslant 1

, then for each integer

q
⩾
1
,

|

Σ

1
⩽
x
⩽
q

exp
⁡
(
2
π
i
f
(
x
)

/

q
)

|

⩽

q

1

/

2

(
D
,
q
)

d
n

(
q
)

q \geqslant 1,|{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)| \leqslant {q^{1/2}}(D,q){d_n}(q)

, provided the discriminant D of the derivative of f does not vanish identically, where

d
n

(
q
)

{d_n}(q)

is the number of representations of q as a product of n factors.
Spectral multiplicity for tensor products of normal operators
Authors: Azoff Edward A. 
Published: June 2010

Two normal operators

N
1

{N_1}

and

N
2

{N_2}

are constructed such that for any pair

m
1

{m_1}

and

m
2

{m_2}

of their respective multiplicity functions, the ’convolution’

(

m
1

∗

m
2

)
(
λ
)
≡
Σ
{

m
1

(

λ
1

)
⋅

m
2

(

λ
2

)

|

λ
1

⋅

λ
2

=
λ
}

({m_1} * {m_2})(\lambda ) \equiv \Sigma \{ {m_1}({\lambda _1}) \cdot {m_2}({\lambda _2})|{\lambda _1} \cdot {\lambda _2} = \lambda \}

fails to be a multiplicity function for the tensor product

N
1

⊗

N
2

{N_1} \otimes {N_2}

.
More on convergence in unitary matrix spaces
Authors: Arazy Jonathan 
Published: June 2010

Let

E
E

be a symmetric sequence space satisfying the Radon-Riesz Property
\[

{

|

|

x
n

|

|

→

|

|

x

|

|

and

x
n

→
x

weakly

}
⇒

|

|

x
n

−
x

|

|

→
0
,

\{ ||{x_n}|| \to ||x||{\text {and }}{x_n} \to x{\text { weakly}}\} \Rightarrow ||{x_n} - x|| \to 0,

\]
then the same is true for the associated unitary matrix space

C

E

C_{E}

.
Derivatives of polynomials with positive coefficients
Authors: Varma A. K. 
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

n
n

with positive coefficients. We set
\[

I
n

=

|

|

P
n
′

(
x
)
ω
(
x
)

|

|

L
2

[
0
,
∞
)

|

|

P
n

(
x
)
ω
(
x
)

|

|

L
2

[
0
,
∞
)

.

{I_n} = \frac {{||P’_n(x)\omega (x)|{|_{{L_2}[0,\infty )}}}}{{||{P_n}(x)\omega (x)|{|_{{L_2}[0,\infty )}}}}.

\]
In this work upper bounds of

I

n

I_{n}

are investigated. We restrict ourselves here with the case

ω
(
x
)
=

x

α

/

2

e

−
x

/

2

\omega (x) = {x^{\alpha /2}}{e^{ - x/2}}

. Reslts are shown to be best possible.
Lipschitz functions and spectral synthesis
Authors: Lee Sung Yung 
Published: June 2010

An

S
S

-set in the circle group

T
T

is a closed subset

S
S

of

T
T

for which

j
(
S
)

¯

=
k
(
S
)

\overline {j(S)} = k(S)

. We construct a non-

S
S

-set

S
S

satisfying
\[

⋃

α
>
0

Lip

α

(
T
)
∩
k
(
S
)
⊂

j
(
S
)

¯

.

\bigcup \limits _{\alpha > 0} {{{\operatorname {Lip} }_\alpha }(T) \cap k(S) \subset \overline {j(S)} .}

\]
Thus

Lip
α

(
T
)
∩
A
(
T
)

{\operatorname {Lip} _\alpha }(T) \cap A(T)

is not a big enough part of

A
(
T
)

A(T)

to test the synthesizability of a given closed subset of

T
T

.
Derivatives of 𝐻^{𝑝} functions
Authors: Øyma Knut , Rookshin Serge 
Published: June 2010

We prove that if

{

z
n

}

\{ {z_n}\}

is uniformly separated and

f
∈

H
p

f \in {H^p}

, then

{

f

(
k
)

(

z
n

)

(
1
−

|

z
n

|

2

)

k
+
1

/

p

}

n
=
1

∞

∈

l
p

for

k
=
1
,
2
,
…

\{ {f^{(k)}}({z_n}){(1 - {\left | {{z_n}} \right |^2})^{k + 1/p}}\} _{n = 1}^\infty \in {l^p}\;{\text {for }}k = 1,2, \ldots

.
The asymptotic behavior of a class of nonlinear differential equations of second order
Authors: Tong Jing Cheng 
Published: June 2010

Let

u
+
f
(
t
,
u
)
=
0

u + f(t,u) = 0

be a nonlinear differential equation. If there are two nonnegative continuous functions

υ
(
t
)

\upsilon (t)

,

φ
(
t
)

\varphi (t)

for

t
⩾
0

t \geqslant 0

, and a continuous function

g
(
u
)

g(u)

for

u
⩾
0

u \geqslant 0

, such that (i)

∫
1
∞

υ
(
t
)
φ
(
t
)

d
t
>
∞

\smallint _1^\infty \upsilon (t)\varphi (t)\;dt > \infty

; (ii) for

u
>
0

u > 0

,

g
(
u
)

g(u)

is positive and nondecreasing; (iii)

|

f
(
t
,
u
)

|

>
υ
(
t
)
φ
(
t
)
g
(

|
u
|

/

t
)

\left | {f(t,u)} \right | > \upsilon (t)\varphi (t)g(\left | u \right |/t)

for

t
⩾
1

t \geqslant 1

,

−
∞
>
u
>
∞

- \infty > u > \infty

, then the equation has solutions asymptotic to

a
+
b
t

a + bt

, where

a
a

,

b
b

are constants and

b
≠
0

b \ne 0

. Our result generalizes a theorem of D. S. Cohen [3].
The capacity of 𝐶₅ and free sets in 𝐶ₘ²
Authors: Mead D. G. , Narkiewicz W. 
Published: June 2010

In a recent paper, S. K. Stein examined the problem of determining the cardinality,

τ
(

C
m
k

)

\tau (C_m^k)

, of the largest subset

S
S

of the direct product

C
m
k

C_m^k

of

k
k

copies of

C
m

{C_m}

such that distinct sums of elements of

S
S

yield distinct elements of

C
m
k

C_m^k

. In this paper we show that

τ
∗

(

C
5

)
=

lim

k
→
∞

(
τ
(

C
5
k

)

/

k
)
=
2

{\tau ^* }({C_5}) = {\lim _{k \to \infty }}(\tau (C_5^k)/k) = 2

, answering a question raised by Stein. We also produce an infinite set of

m
m

’s such that

τ
(

C
m
2

)
>
2
[

log
2

m
]

\tau (C_m^2) > 2[{\log _2}m]

.
Free 𝐸-𝑚 groups and free 𝐸-𝑚 semigroups
Authors: Tamura Takayuki 
Published: June 2010

A group [semigroup] is called

E
E

-

m
m

if it satisfies the identity

(
x
y

)
m

=

x
m

y
m

{(xy)^m} = {x^m}{y^m}

. In this paper the author studies a necessary and sufficient condition on

m
m

and

n
n

for the free

E
E

-

m
m

group [semigroup] to be homomorphic onto the free

E

-

n

E{\text { - }}n

group [semigroup].
Finite homological dimension of 𝐵𝑃_{∗}(𝑋) for infinite complexes
Authors: Landweber Peter S. 
Published: June 2010

The main result proved here is that

B

P

∗

(
E
G

×
G

X
)

{\text {B}}{{\text {P}}_* }(EG{ \times _G}X)

has finite homological dimension when

G
=

Z

p

G = {{\mathbf {Z}}_p}

and

X
X

is a finite

G

- CW

G{\text { - CW}}

-complex. The argument uses

B

P

∗

BP

{\text {B}}{{\text {P}}_*}{\text {BP}}

-comodules.
Difference equations and multipoint boundary value problems
Authors: Eloe P. W. 
Published: June 2010

Let

I
=

{

a
,
a
+
1
,
…
,
b

}

I = \left \{ {a,a + 1, \ldots ,b} \right \}

be finite, let

n
⩾
1

n \geqslant 1

, and let

I
j

=

{

a
,
a
+
1
,
…
,
b
+
j

}

{I^j} = \left \{ {a,a + 1, \ldots ,b + j} \right \}

,

j
=
1
,
…
,
n

j = 1, \ldots ,n

. Let

B
B

be the set of mappings from

I
n

{I^n}

into the reals and define the linear difference operator

P
P

by (1)
\[

P
u
(
m
)
=

∑

j
=
0

n

α
j

(
m
)
u
(
m
+
j
)
,

where

m
∈
I
,

α
n

(
m
)
≡
1
,

and

α
0

(
m
)
≠
0

on

I
.

Pu(m) = \sum \limits _{j = 0}^n {{\alpha _j}(m)u(m + j),} \quad {\text {where }}m \in I,{\alpha _n}(m) \equiv 1,{\text {and }}{\alpha _0}(m) \ne 0{\text { on }}I.

\]
Existence of solutions theorems and iteration schemes that approximate solutions are given for boundary value problems of the form

P
u
(
m
)
=
f
(
m
,
u
,
E
u
,
…
,

E

n
−
1

u
)

Pu(m) = f(m,u,Eu, \ldots ,{E^{n - 1}}u)

, with boundary conditions

T
u
(
m
)
=
r

Tu(m) = r

, where

P
P

is defined by (1),

E
j

u
(
m
)
=
u
(
m
+
j
)

{E^j}u(m) = u(m + j)

,

j
=
0
,
1
,
…
,
n
−
1

j = 0,1, \ldots ,n - 1

,

f
:
I
×

R

n

→

R

f:I \times {{\mathbf {R}}^n} \to {\mathbf {R}}

is continuous, and

T
:
B
→

R

n

T:B \to {{\mathbf {R}}^n}

is a continuous linear operator. The results are based on solutions of difference inequalities and sign properties of associated Green’s functions.
Relative normal complements in finite groups
Authors: Ferguson Pamela A. 
Published: June 2010

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

denotes the following configuration:

H
H

and

H
0

{H_0}

are the subgroups of the finite group

G
G

with

H
0

⊴
H

{H_0} \trianglelefteq H

is the set of primes dividing

(
H
:

H
0

)

(H:{H_0})

. For

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

we consider conditions

(

A

)

({\text {A}})

,

(

B

0

)

({{\text {B}}_0})

, and

(

C

)

({\text {C}})

:

(

A

)

({\text {A}})

Any two

π
\pi

-elements of

H
−

H
0

H - {H_0}

which are

G
G

-conjugate are

H
H

-conjugate.

(

B

0

)

({{\text {B}}_0})

For each

π
\pi

-element

x
∈
H
−

H
0

x \in H - {H_0}

,

C
G

(
x
)
=
I
(
x
)

C
H

(
x
)

{C_G}(x) = I(x){C_H}(x)

where

I
(
x
)

I(x)

is a normal

π
′

\pi ’

-subgroup of

C
G

(
x
)

{C_G}(x)

.

(

C

)

|

(
H
−

H
0

)

G
,
π

|

=
(
G
:
H
)

|

H
−

H
0

|

({\text {C}})\left | {{{(H - {H_0})}^{G,\pi }}} \right | = (G:H)\left | {H - {H_0}} \right |

. We show that if

(
G
,
H
,

H
0

,
π
)

(G,H,{H_0},\pi )

satisfies

(

B

0

)

({{\text {B}}_0})

and

(

C

)

({\text {C}})

, or

(

A

)

({\text {A}})

and

(

B

0

)

({{\text {B}}_0})

, and if

H

/

H
0

H/{H_0}

is solvable, then there is a unique relative normal complement

G
0

{G_0}

of

H
H

over

H
0

{H_0}

.
Some inequalities of algebraic polynomials having all zeros inside [-1,1]
Authors: Varma A. K. 
Published: June 2010

Let

H
n

{H_n}

be the set of all algebraic polynomials whose degree is

n
n

and whose zero are all real and lie inside

[
−
1
,
1
]

[ - 1,1]

. Then for

n
n

even we have

(
n
=
2
m
)

(n = 2m)

\[

∫

−
1

1

(

P
n

(
x
)
)

2

⩾
(
n

/

2
+
3

/

4
+
3

/

4
(
n
−
1
)
)

∫

−
1

1

P
n
2

(
x
)
d
x

,

\int _{ - 1}^1 {{{({P_n}(x))}^2} \geqslant (n/2 + 3/4 + 3/4(n - 1))\int _{ - 1}^1 {P_n^2(x)dx} } ,

\]
where equality holds iff

P
n

(
x
)
=

(
1
−

x
2

)
m

{P_n}(x) = {(1 - {x^2})^m}

. If

n
n

is an odd positive integer, a similar inequality is valid (see (1.6) below). In the case

P
n

∈

H
n

{P_n} \in {H_n}

and subject to the condition

P
n

(
1
)
=
1

{P_n}(1) = 1

, then
\[

∫

−
1

1

(

P
′

n

(
x
)

)
2

d
x
⩾

n
4

+

1
8

+

1

8
(
2
n
−
1
)

,

\int _{ - 1}^1 {({{P’}_n}} (x){)^2}dx \geqslant \frac {n}{4} + \frac {1}{8} + \frac {1}{{8(2n - 1)}},

\]
, where equality holds for

P
n

(
x
)
=

(
(
1
+
x
)

/

2

)
n

{P_n}(x) = {((1 + x)/2)^n}

.