bluuebunny/crossref_metadata_2025_split huggingface dataset:
Hall subgroups and 𝑝-solvability
Authors: Gonçalves A. , Ho C. Y. 
Published: June 2010

Let

G
G

be a finite group and let

π
(
G
)
=
{
p
,

q
1

,
…
,

q
r

}

\pi (G) = \{ p,{q_1}, \ldots ,{q_r}\}

be the set of all prime divisors of

G
G

. Suppose there is a

p
′

p’

-Hall subgroup

H
H

. If there are subgroups

P
,

Q
1

,
…
,

Q
r

P,{Q_1}, \ldots ,{Q_r}

such that

P
ϵ

Syl
p

(
G
)
,

Q
i

ϵ

Syl

q
i

(
H
)

P\epsilon {\operatorname {Syl} _p}(G),{Q_i}\epsilon {\operatorname {Syl} _{{q_i}}}(H)

, and

L
i

=
P

Q
i

{L_i} = P{Q_i}

is a subgroup,

i
=
1
,
…
,
r

i = 1, \ldots ,r

, then

G
G

is

p
p

-solvable. Moreover, if the subgroup

H
H

is solvable, then

G
G

is solvable too.
Higher derivations on finitely generated integral domains. II
Authors: Brown William C. 
Published: June 2010

We prove Theorem. Let

A
=
k
[

x
1

,
…
,

x
m

]

A = k[{x_1}, \ldots ,{x_m}]

be a finitely generated integral domain over a field

k
k

of characteristic zero. Then

A
A

regular, i.e. the local ring

A
q

{A_q}

is regular for every prime ideal

q
⊆
A

q \subseteq A

, is equivalent to the following two conditions: (1) no prime of

A
A

of height greater than one is differential, and (2) for all

ϕ
∈

Hom
k

(
A
,
A
)
,
ϕ
∈

Der
k
n

⁡
(
A
)

\phi \in {\operatorname {Hom} _k}(A,A),\phi \in \operatorname {Der} _k^n(A)

if and only if

Δ
ϕ
∈

Σ

i
=
1

n
−
1

Der
k
i

⁡
(
A
)
∪

Der
k

n
−
i

⁡
(
A
)
(
n
=
1
,
2
,
…
)

\Delta \phi \in \Sigma _{i = 1}^{n - 1}\operatorname {Der} _k^i(A) \cup \operatorname {Der} _k^{n - i}(A)(n = 1,2, \ldots )

. Here

Δ
\Delta

denotes the Hochschild coboundary operator,

∪
\cup

denotes the cup product, and

Der
k
n

⁡
(
R
)

\operatorname {Der} _k^n(R)

is the module of higher derivations of rank

n
n

.
A sufficient condition for eventual disconjugacy
Authors: Trench William F. 
Published: June 2010

It is known that the scalar equation

y

(
n
)

+

p
1

(
t
)

y

(
n
−
1
)

+
⋯
+

p
n

(
t
)
y
=
0
,
t
>
0
,
n
>
1

{y^{(n)}} + {p_1}(t){y^{(n - 1)}} + \cdots + {p_n}(t)y = 0,t > 0,n > 1

, is eventually disconjugate if

p
1

,
…
,

p
n

ϵ
C
[
0
,
∞
)

{p_1}, \ldots ,{p_n}\epsilon C[0,\infty )

and

∫

∞

|

p
i

(
t
)

|

t

i
−
1

d
t
>
∞
,
1
⩽
i
⩽
n

\int {^\infty |{p_i}(t)|{t^{i - 1}}dt > \infty ,1 \leqslant i \leqslant n}

. This paper presents a weaker integral condition which also implies that the given equation is eventually disconjugate.
Oscillation properties of perturbed disconjugate equations
Authors: Trench William F. 
Published: June 2010

Oscillation conditions are given for the equation

L
u

+
f
(
t
,
u
)
=
0

{L_u} + f(t,u) = 0

, where
\[

L
u
=

1

β
n

d

d
t

1

β

n
−
1

⋯

d

d
t

1

β
1

d

d
t

u

β
0

(
n
⩾
2
)
,

Lu = \frac {1} {{{\beta _n}}}\frac {d} {{dt}}\frac {1} {{{\beta _{n - 1}}}} \cdots \frac {d} {{dt}}\frac {1} {{{\beta _1}}}\frac {d} {{dt}}\frac {u} {{{\beta _0}}}(n \geqslant 2),

\]
with

β
0

,
…
,

β
n

{\beta _0}, \ldots ,{\beta _n}

positive and continuous on

(
0
,
∞
)
,
∫

∞

β
i

d
t
=
∞
(
1
⩽
i
⩽
n
−
1
)

(0,\infty ),\int {^\infty {\beta _i}dt = \infty (1 \leqslant i \leqslant n - 1)}

, and

f
f

subject to conditions which include

u
f
(
t
,
u
)
⩾
0

uf(t,u) \geqslant 0

. The results obtained include previously known oscillation conditions for the equation

u

(
n
)

+
f
(
t
,
u
)
=
0

{u^{(n)}} + f(t,u) = 0

for both linear and nonlinear cases.
The maps 𝐵𝑆𝑝(1)→𝐵𝑆𝑝(𝑛)
Authors: Mahmud Zafer 
Published: June 2010

Let

S
p
(
n
)

Sp(n)

be the symplectic Lie group. Then it is known that given a map

f
:
B
S
p
(
1
)
→
B
S
p
(
1
)
,

f

∗

:

H
4

(
B
S
p
(
1
)
,

Z

)
→

H
4

(
B
S
p
(
1
)
,

Z

)

f:BSp(1) \to BSp(1),{f^{\ast }}:{H^4}(BSp(1),{\mathbf {Z}}) \to {H^4}(BSp(1),{\mathbf {Z}})

is zero or multiplication by the square of an odd integer. We generalise the latter part of this result using symplectic

K

∗

{K^{\ast }}

-theory.
A class of 𝐿^{𝑝}-bounded pseudo-differential operators
Authors: Illner Reinhard 
Published: June 2010

Pseudo-differential operators with symbol

p
(
x
,
ξ
,
y
)
∈

S

ρ
,
δ
,
ε

μ

,
μ
≤
(
ρ
−
1
)
(
n
+
1
)

p(x,\xi ,y) \in S_{\rho ,\delta ,\varepsilon }^\mu ,\mu \leq (\rho - 1)(n + 1)

, are proven to be generally

L
p

{L^p}

-bounded for

1
>
p
>
∞

1 > p > \infty

.
On the characterization of Souslin and Borel sets
Authors: Reiss R.-D. 
Published: June 2010

Let

X

\mathfrak {X}

be an arbitrary family of sets in the basic set

X
X

. In this paper Souslin-

X

\mathfrak {X}

sets are represented as projections on

X
X

of certain

σ
δ

\sigma \delta

-sets in the cartesian product space of

X
X

and the Baire space (or

X
X

and the real line). For

L

-

X

L{\text { - }}\mathfrak {X}

sets (i.e. Souslin sets defined with disjoint unions; ensembles d’unicité) injective projections are considered. The results also apply to Borel-

X

\mathfrak {X}

sets since the system of

L

-

X

L{\text { - }}\mathfrak {X}

sets includes the Borel-

X

\mathfrak {X}

sets under suitable conditions.
Intersections of commutants of analytic Toeplitz operators
Authors: Thomson James E. 
Published: June 2010

In this paper we study the intersection of commutants of analytic Toeplitz operators. Our main result is that if

ϕ
\phi

is a finite Blaschke product and

Ψ
ϵ

H
∞

\Psi \epsilon {H^\infty }

, then

{

T
ϕ

}
′

∩
{

T
Ψ

}
′

=
{

T
I

}
′

\{ {T_\phi }\} ’ \cap \{ {T_\Psi }\} ’ = \{ {T_I}\} ’

where

I
I

is a finite Blaschke product and

ϕ
\phi

and

Ψ
\Psi

are functions of

I
I

. The key step is a function-theoretic theorem describing the relationship between a finite Blaschke product and any

H
∞

{H^\infty }

function.
Approximation to fixed points of generalized nonexpansive mappings
Authors: Wong Chi Song 
Published: June 2010

Let

K
K

be a convex subset of a uniformly convex Banach space. It is proved that if

K
K

is compact, then the fixed points of a continuous generalized nonexpansive self-mapping

T
T

on

K
K

can be approximated by the iterates of

T
t

{T_t}

with

t
∈
(
0
,
1
)
,

T
t

(
x
)
=
(
1
−
t
)
x
+
t
T
(
x
)
,
x
∈
K
;

T
t

t \in (0,1),{T_t}(x) = (1 - t)x + tT(x),x \in K;{T_t}

is asymptotically regular if

T
T

has a fixed point.
A note on generalizing alternative rings
Authors: Hentzel Irvin Roy , Piacentini Cattaneo Giulia Maria 
Published: June 2010

Let

R
R

be a nonassociative ring of characteristic different from

2
2

and

3
3

which satisfies the following identities:
\[

(

i)

(
a
b
,
c
,
d
)
+
(
a
,
b
,
[
c
,
d
]
)
=
a
(
b
,
c
,
d
)
+
(
a
,
c
,
d
)
b
,

({\text {i)}}\;(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b,

\]

\[

(

ii)

(
a
,
a
,
a
)
=
0
,

({\text {ii)}}\;(a,a,a) = 0,

\]

\[

(

iii)

(
a
,
b
∘
c
,
d
)
=
b
∘
(
a
,
c
,
d
)
+
c
∘
(
a
,
b
,
d
)

({\text {iii)}}\;(a,b \circ c,d) = b \circ (a,c,d) + c \circ (a,b,d)

\]
for all

a
,
b
,
c
,
d
∈
R

a,b,c,d \in R

and with

x
∘
y
=
(
x
y
+
y
x
)

/

2

x \circ y = (xy + yx)/2

. We prove that if

R
R

is semiprime, then

R
R

is alternative.
On 𝐿¹ convergence of certain cosine sums
Authors: Garrett John W. , Stanojević Časlav V. 
Published: June 2010

Rees and Stanojević introduced a new class of modified cosine sums

{

g
n

(
x
)
=

1
2

∑

k
=
0

n

Δ
a
(
k
)
+

∑

k
=
1

n

∑

j
=
k

n

Δ
a
(
j
)
cos
⁡
k
x
}

\{ {g_n}(x) = \tfrac {1} {2}\sum \nolimits _{k = 0}^n {\Delta a(k) + \sum \nolimits _{k = 1}^n {\sum \nolimits _{j = k}^n {\Delta a(j)\cos kx\} } } }

and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series

f
f

with coefficients of bounded variation, a Rees-Stanojević cosine sum

g
n

{g_n}

can be associated such that

g
n

{g_n}

converges to

f
f

pointwise, and a necessary and sufficient condition for

L
1

{L^1}

convergence of

g
n

{g_n}

to

f
f

is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions.
An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside [-1,+1]
Authors: Varma A. K. 
Published: June 2010

Let

P
n

(
x
)

{P_n}(x)

be an algebraic polynomial of degree

⩽
n

\leqslant n

having all its zeros inside

[
−
1
,
+
1
]

[ - 1, + 1]

; then we have
\[

∫

−
1

1

P
n

′

2

(
x
)
d
x
>
(
n

/

2
)

∫

−
1

1

P
n
2

(
x
)
d
x
.

\int _{ - 1}^1 {P_n^{’2}(x)dx > (n/2)\int _{ - 1}^1 {P_n^2(x)dx.} }

\]
The result is essentially best possible. Other related results are also proved.
Idempotents of norm one and Banach algebra representations of compact groups
Authors: Kalton N. J. , Wood G. V. 
Published: June 2010

Let

G
G

be a finite group of order

n
n

and let

A
A

be a (real or complex) Banach algebra. Rudin and Schneider [3] ask whether a mapping

f
:
G
→
A

f:G \to A

satisfying

|

|

f
(
x
)

|

|

=
1

||f(x)|| = 1

and

f
(
x
)
=
(
1

/

n
)

Σ

y
∈
G

f
(
x

y

−
1

)
f
(
y
)

f(x) = (1/n){\Sigma _{y \in G}}f(x{y^{ - 1}})f(y)

is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if

A
A

is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and at the same time prove the natural generalization to compact groups. This allows us to characterize norm one idempotents in generalized group algebras.
Existence of inverses and square roots in locally Banach semigroups with identity
Authors: Eslinger Robert C. 
Published: June 2010

Let

S
S

be a multiplicative topological semigroup with identity

e
e

. Suppose

D
D

is an open subset containing

e
e

and

h
h

is a homeomorphism from

D
D

onto a Banach space

B
B

with

h
(
e
)
=
0

h(e) = 0

. Define the function

P
P

by

P
(
x
,
y
)
=
h
[

h

−
1

(
x
)
⋅

h

−
1

(
y
)
]

P(x,y) = h[{h^{ - 1}}(x) \cdot {h^{ - 1}}(y)]

. A new implicit function theorem is applied to the function

P
P

to show the existence of inverses and square roots of elements in a neighborhood of the identity. It is assumed that

P
P

satisfies the following condition: There exist a one-one function

A
A

from a subset of

B
B

into

B
B

and positive numbers

r
,
M

r,M

, and

c
c

such that (i) if

|

|

x

|

|

>
r

||x|| > r

then

x
∈
dom
⁡
(

A

−
1

)

x \in \operatorname {dom} ({A^{ - 1}})

and

|

|

A

−
1

(
x
)

|

|

⩽
M

|

|

x

|

|

||{A^{ - 1}}(x)|| \leqslant M||x||

, (ii)

c
M
>
1

cM > 1

, and (iii) if

|

|

x
i

|

|

,

|

|

y
i

|

|

>
r
(
i
=
1
,
2
)

||{x_i}||,||{y_i}|| > r(i = 1,2)

then

(

x
i

,

y
j

)
∈
dom
⁡
(
P
)
(
i
,
j
=
1
,
2
)

({x_i},{y_j}) \in \operatorname {dom} (P)(i,j = 1,2)

,
\[

|

|

P
(

x
1

,

y
1

)
−
P
(

x
2

,

y
2

)
−
A
(

y
1

−

y
2

)

|

|

⩽
c

|

|

y
1

−

y
2

|

|

,

||P({x_1},{y_1}) - P({x_2},{y_2}) - A({y_1} - {y_2})|| \leqslant c||{y_1} - {y_2}||,

\]
and
\[

|

|

P
(

x
1

,

y
1

)
−
P
(

x
2

,

y
1

)
−
A
(

x
1

−

x
2

)

|

|

⩽
c

|

|

x
1

−

x
2

|

|

.

||P({x_1},{y_1}) - P({x_2},{y_1}) - A({x_1} - {x_2})|| \leqslant c||{x_1} - {x_2}||.

\]
Weak convergence of semigroups implies strong convergence of weighted averages
Authors: Fong Humphrey 
Published: June 2010

For a fixed

p
,
1
⩽
p
>
∞

p,1 \leqslant p > \infty

, let

{

T
t

:
t
>
0
}

\{ {T_t}:t > 0\}

be a strongly continuous semigroup of positive contractions on

L
p

{L_p}

of a

σ
\sigma

-finite measure space. We show that weak convergence of

{

T
t

:
t
>
0
}

\{ {T_t}:t > 0\}

in

L
p

{L_p}

is equivalent with the strong convergence of the weighted averages

∫
0
∞

T
t

f

μ
n

(
d
t
)
(
n
→
∞
)

\int _0^\infty {{T_t}f{\mu _n}(dt)(n \to \infty )}

for every

f
∈

L
p

f \in {L_p}

and every sequence

(

μ
n

)

({\mu _n})

of signed measures on

(
0
,
∞
)

(0,\infty )

, satisfying

sup
n

|

|

μ
n

|

|

>
∞
;

lim
n

μ
n

(
0
,
∞
)
=
1

{\sup _n}||{\mu _n}|| > \infty ;{\lim _n}{\mu _n}(0,\infty ) = 1

; and for each

d
>
0
,

lim
n

sup

c
⩾
0

|

μ
n

|

(
c
,
c
+
d
]
=
0

d > 0,{\lim _n}{\sup _{c \geqslant 0}}|{\mu _n}|(c,c + d] = 0

. The positivity assumption is not needed if

p
=
1

p = 1

or 2. We show that such a result can be deduced-not only in

L
p

{L_p}

, but in general Banach spaces-from the corresponding discrete parameter version of the theorem.
Ergodic theorems of weak mixing type
Authors: Jones Lee K. , Lin Michael 
Published: June 2010

Given a linear contraction

T
T

on a Banach space

X
X

and

x
∈
X

x \in X

, the convergence
\[

∀

x
∗

∈

X
∗

N

−
1

∑

i
=
1

N

|

⟨

x
∗

,

T
i

x
⟩

|

→

N
→
∞

0

\forall {x^ \ast } \in {X^ \ast }{N^{ - 1}}\sum \limits _{i = 1}^N {|\langle {x^ \ast },{T^i}x\rangle |} \xrightarrow [{N \to \infty }]{}0

\]
is shown to be equivalent to the convergence
\[

sup

|

|

x
∗

|

|

⩽
1

N

−
1

∑

j
=
1

N

|

⟨

x
∗

,

T

k
j

x
⟩

|

→
0

\sup \limits _{||{x^ \ast }|| \leqslant 1} {N^{ - 1}}\sum \limits _{j = 1}^N {|\langle {x^ \ast },{T^{{k_j}}}x\rangle | \to 0}

\]
for every subsequence with

k
j

/

j

{k_j}/j

bounded. A sufficient condition is that, for some

{

n
i

}
,

T

n
i

x
→
0

\{ {n_i}\} ,{T^{{n_i}}}x \to 0

weakly.
Some remarks on summability factors
Authors: Gavin Lloyd A. 
Published: June 2010

Bosanquet [2] showed that a necessary and sufficient condition for

Σ

k
=
1

∞

x
k

y
k

\Sigma _{k = 1}^\infty {x_k}{y_k}

to be Cesàro summable of order

n
n

(

n
n

is a nonnegative integer) whenever

σ
k
n

(
y
)
=
o
(
k
)

\sigma _k^n(y) = o(k)

where

σ
k
n

(
y
)

\sigma _k^n(y)

is the

k
k

th Cesàro mean of

y
y

of order

n
n

is that

Σ

k
=
1

∞

k

n
+
1

|

Δ

n
+
1

x
k

|

>
∞

\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty

and

lim

k
→
0

k

x
k

=
0

{\lim _{k \to 0}}k{x_k} = 0

. The main result of this paper is to show that a necessary and sufficient condition for

Σ

k
=
1

∞

x
k

y
k

\Sigma _{k = 1}^\infty {x_k}{y_k}

to be Cesàro summable of order

n
n

(

n
n

is a nonnegative integer) whenever

Σ

k
=
1

∞

k

n
+
1

|

Δ

n
+
1

x
k

|

>
∞

\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| > \infty

and

lim

k
→
∞

k

x
k

=
0

{\lim _{k \to \infty }}k{x_k} = 0

is that

σ
k
n

(
y
)
=
o
(
k
)

\sigma _k^n(y) = o(k)

.
A remark on the strong law of large numbers
Authors: Chen Robert 
Published: June 2010

Let

X
1

,

X
2

,
…

{X_1},{X_2}, \ldots

be mutually independent random variables such that

E
(

X
n

)
=
0

E({X_n}) = 0

and

E
(

X
n
2

)
=

σ
n
2

=
1

E(X_n^2) = \sigma _n^2 = 1

for all

n
=
1
,
2
,
…

n = 1,2, \ldots

. For each

n
=
1
,
2
,
…

n = 1,2, \ldots

let

S
n

=

∑

j
=
1

n

X
j

{S_n} = \sum \nolimits _{j = 1}^n {{X_j}}

; then, by the Kolmogorov criterion for mutually independent random variables,

S
n

/

n

1

/

2
+
α

→
0

{S_n}/{n^{1/2 + \alpha }} \to 0

almost surely as

n
→
∞

n \to \infty

for any positive constant

α
\alpha

. A deeper understanding of this theorem will be facilitated if we know the order of magnitude of

E
{

N
∞

(
α
,
ε
)
}

E\{ {N_\infty }(\alpha ,\varepsilon )\}

as

ε
→

0
+

\varepsilon \to {0^ + }

, where

N
∞

(
α
,
ε
)

{N_\infty }(\alpha ,\varepsilon )

is the integer-valued random variable defined by

N
∞

(
α
,
ε
)
=

∑

n
=
1

∞

χ

(
ε

n

1

/

2
+
α

,
∞
)

(

|

S
n

|

)

{N_\infty }(\alpha ,\varepsilon ) = \sum \nolimits _{n = 1}^\infty {{\chi _{(\varepsilon {n^{1/2 + \alpha }},\infty )}}(|{S_n}|)}

. The present note does the work for a wide class of random variables by using Esseen’s theorem and Katz-Petrov’s theorem.
On perturbation of unstable second order linear differential equations
Authors: Hatvani L. , Pintér L. 
Published: June 2010

In connection with a conjecture of J. M. Bownds, conditions will be given on the fundamental system of the solutions of the unstable differential equation

y
+
a
(
t
)
y
=
0

y + a(t)y = 0

which assure that the differential equation

x
+
a
(
t
)
x
=
g
(
t
,
x
,

x
′

)

x + a(t)x = g(t,x,x’)

has a solution with the property
\[

lim
sup
(

|

x
(
t
)

|

+

|

x
′

(
t
)

|

)
=
∞

as

t
→
∞
,

\lim \sup (|x(t)| + |x’(t)|) = \infty \quad {\text {as }}t \to \infty ,

\]
provided that

g
(
t
,
x
,

x
′

)

g(t,x,x’)

is “sufficiently small".
On mappings contractive in the sense of Kannan
Authors: Janos Ludvik 
Published: June 2010

Let

f
:
X
→
X

f:X \to X

be a continuous compact mapping of a metric space

(
X
,
d
)

(X,d)

into itself with the property that

x
,
y
∈
X

x,y \in X

and

x
≠
y

x \ne y

implies

d
(
f
(
x
)
,
f
(
y
)
)
>

1
2

[
d
(
x
,
f
(
x
)
)
+
d
(
y
,
f
(
y
)
)
]

d(f(x),f(y)) > \tfrac {1} {2}[d(x,f(x)) + d(y,f(y))]

. It is shown that under these conditions

f
f

has a unique fixed point and, moreover,

f
f

is a Banach contraction relative to a suitable remetrization of the space

X
X

. A similar result concerning condensing mappings is also obtained.
On maximality of Gorenstein sequences
Authors: Marinari Maria Grazia 
Published: June 2010

It is well known that if

A
A

is a Gorenstein ring, then every ideal generated by a regular sequence

x

⊂
A

{\text {x}} \subset A

has irreducible (minimal) primary components. This feature led us to define a Gorenstein sequence of a ring

A
A

to be any ordered regular sequence

x

=
{

x
1

,
…
,

x
r

}
⊂
A

{\text {x}} = \{ {x_1}, \ldots ,{x_r}\} \subset A

such that for every

i
∈
{
1
,
…
,
r
}

i \in \{ 1, \ldots ,r\}

the ideal

(

x
1

,
…
,

x
i

)

({x_1}, \ldots ,{x_i})

has irreducible minimal primary components. We showed for Gorenstein sequences (

G

{\mathbf {G}}

-sequences for short) some parallels of well-known properties of regular sequences and moreover by means of

G

{\mathbf {G}}

-sequences we gave the following natural characterization of local Gorenstein rings: “A local ring

(
A
,

m

)

(A,\;\mathfrak {m})

is Gorenstein iff

m

\mathfrak {m}

contains a

G

- sequence of length =

K

-

dim
⁡
A

{\mathbf {G}}{\text { - sequence of length = }}K{\text { - }}\dim A

". In this note we are going to give some information about “maximality” of

G

{\mathbf {G}}

-sequences in a local ring

A
A

, producing sufficient conditions on

A
A

in order that the maximal

G

{\mathbf {G}}

-sequences of

A
A

all have the same length, i.e. in order to give a “good” definition of

G

{\mathbf {G}}

-depth

A
A

. Furthermore, we will state some results about the

G

{\mathbf {G}}

-depth behavior with respect to local flat ring homomorphisms.
An existence theorem for boundary value problems of nonlinear ordinary differential equations
Authors: Klaasen Gene A. 
Published: June 2010

Let

f
f

be continuous on

(
a
,
b
)
×

R
n

(a,b) \times {R^n}

and suppose solutions of initial value problems for

y

(
n
)

=
f
(
t
,
y
,
…
,

y

(
n
−
1
)

)

{y^{(n)}} = f(t,y, \ldots ,{y^{(n - 1)}})

exist on

(
a
,
b
)

(a,b)

. Relaxing the assumption that solutions of initial value problems are unique, global existence of solutions of the boundary value problem
\[

y

(
n
)

=
f
(
t
,
y
,
…
,

y

(
n
−
1
)

)
,
y
(

t
i

)
=

α
i

for

1
⩽
i
⩽
n
,

{y^{(n)}} = f(t,y, \ldots ,{y^{(n - 1)}}),y({t_i}) = {\alpha _i}\quad {\text {for }}1 \leqslant i \leqslant n,

\]
is established assuming uniqueness of solutions of these problems and a compactness property of solutions of the differential equation.
New subclasses of the class of close-to-convex functions
Authors: Chichra Pran Nath 
Published: June 2010

In this paper we introduce new subclasses of the class of close-to-convex functions. We call a regular function

f
(
z
)

f(z)

an alpha-close-to-convex function if

(
f
(
z
)

f
′

(
z
)

/

z
)
≠
0

(f(z)f’(z)/z) \ne 0

for z in E and if for some nonnegative real number

α
\alpha

there exists a starlike function

ϕ
(
z
)
=
z
+
⋯

\phi (z) = z + \cdots

such that
\[

Re

[

(
1
−
α
)

z

f
′

(
z
)

ϕ
(
z
)

+
α

(
z

f
′

(
z
)

)
′

ϕ
′

(
z
)

]

>
0

\operatorname {Re} \;\left [ {(1 - \alpha )\frac {{zf’(z)}}{{\phi (z)}} + \alpha \frac {{(zf’(z))’}}{{\phi ’(z)}}} \right ] > 0

\]
for z in E. We have proved that all alpha-close-to-convex functions are close-to-convex and have obtained a few coefficient inequalities for

α
\alpha

-close-to-convex functions and an integral formula for constructing these functions. Let

F

α

{\mathfrak {F}_\alpha }

be the class of regular and normalised functions

f
(
z
)

f(z)

which satisfy

Re

(

f
′

(
z
)
+
α
z
f
(
z
)
)
>
0

\operatorname {Re} \;(f’(z) + \alpha zf(z)) > 0

for z in E.

f
(
z
)
∈

F

α

f(z) \in {\mathfrak {F}_\alpha }

gives

Re
⁡

f
′

(
z
)
>
0

\operatorname {Re} f’(z) > 0

for z in E provided

Re
⁡
α
⩾
0

\operatorname {Re} \alpha \geqslant 0

. A sharp radius of univalence of the class of functions

f
(
z
)

f(z)

for which

z

f
′

(
z
)
∈

F

α

zf’(z) \in {\mathfrak {F}_\alpha }

has also been obtained.
On the Schnirelmann density of the 𝑘-free integers
Authors: Diananda P. H. , Subbarao M. V. 
Published: June 2010

Let

Q
k

(
n
)

{Q_k}(n)

be the number of k-free integers

⩽
n

\leqslant n

and

d
(

Q
k

)

d({Q_k})

the Schnirelmann density of the k-free integers. If

k
⩾
5

k \geqslant 5

, it is shown that

Q
k

(
n
)

/

n
=
d
(

Q
k

)

{Q_k}(n)/n = d({Q_k})

for some n satisfying

6
k

/

2
⩽
n
>

6
k

{6^k}/2 \leqslant n > {6^k}

and certain other properties, and that
\[

d
(

Q
k

)
⩾
1
−

2

−
k

−

3

−
k

−

5

−
k

+
(

3

−
k

+
2
⋅

5

−
k

)

(

6
k

−

3
k

+
1

)

−
1

.

d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}.

\]

d
(

Q
k

)

d({Q_k})

and the n for which

Q
k

(
n
)

/

n
=
d
(

Q
k

)

{Q_k}(n)/n = d({Q_k})

are found for

7
⩽
k
⩽
12

7 \leqslant k \leqslant 12

.
An extension of Carlitz’s bipartition identity
Authors: Andrews George E. 
Published: June 2010

Carlitz’s bipartition identity is extended to a multipartite partition identity by the introduction of the summatory maximum function:
\[

smax
⁡
(

n
1

,

n
2

,
…
,

n
r

)
=

n
1

+

n
2

+
⋯
+

n
r

−
(
r
−
1
)
min
(

n
1

,

n
2

,
…
,

n
r

)
.

\operatorname {smax}(n_1, n_2, \ldots , n_r) = n_1 + n_2 + \cdots + n_r - (r - 1)\min (n_1, n_2, \ldots , n_r).

\]
Let

π
0

(

n
1

,

n
2

,
…
,

n
r

)

\pi _0(n_1, n_2, \ldots , n_r)

denote the number of partitions of

(

n
1

,

n
2

,
…
,

n
r

)

({n_1},{n_2}, \ldots ,{n_r})

in which the minimum coordinate of each part is not less then the summatory maximum of the next part. Let

π
1

(

n
1

,

n
2

,
…
,

n
r

)

{\pi _1}({n_1},{n_2}, \ldots ,{n_r})

denote the number of partitions of

(

n
1

,

n
2

,
…
,

n
r

)

({n_1},{n_2}, \ldots ,{n_r})

in which each part has one of the

2
r
−
1

2r - 1

forms:

(
a
+
1
,
a
,
a
,
…
,
a
)
,
(
a
,
a
+
1
,
a
,
…
,
a
)
,
…
,
(
a
,
a
,
a
,
…
,
a
+
1
)
,
(
r
a
+
2
,
r
a
+
2
,
…
,
r
a
+
2
)
,
(
r
a
+
3
,
r
a
+
3
,
…
,
r
a
+
3
)
,
…
,
(
r
a
+
r
,
r
a
+
r
,
…
,
r
a
+
r
)

(a + 1,a,a, \ldots ,a),(a,a + 1,a, \ldots ,a), \ldots ,(a,a,a, \ldots ,a + 1),(ra + 2,ra + 2, \ldots ,ra + 2),(ra + 3,ra + 3, \ldots ,ra + 3), \ldots ,(ra + r,ra + r, \ldots ,ra + r)

. Theorem:

π
0

(

n
1

,
…
,

n
r

)
=

π
1

(

n
1

,
…
,

n
r

)

{\pi _0}({n_1}, \ldots ,{n_r}) = {\pi _1}({n_1}, \ldots ,{n_r})

.
On certain weighted partitions and finite semisimple rings
Authors: Richmond L. B. , Subbarao M. V. 
Published: June 2010

Let k be a fixed integer

⩾
1

\geqslant 1

and define

τ
k

(
n
)
=

Σ

d
k

/

n

1

{\tau _k}(n) = {\Sigma _{{d^k}/n}}1

. Thus

τ

1

(
n
)

{\tau _{1}}(n)

is the ordinary divisor function and

τ
k

(
n
)

{\tau _k}(n)

is the number of kth powers dividing n. We derive the asymptotic behaviour as

n
→
∞

n \to \infty

of

P
k

(
n
)

{P_k}(n)

defined by
\[

∑

n
=
0

∞

P
k

(
n
)

x
n

=

∏

n
=
1

∞

(
1
−

x
n

)

−

τ
k

(
n
)

.

\sum \limits _{n = 0}^\infty {{P_k}(n){x^n} = \prod \limits _{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .}

\]
Thus

P
k

(
n
)

{P_k}(n)

is the number of partitions of n where we recognize

τ
k

(
m
)

{\tau _k}(m)

different colours of the integer m when it occurs as a summand in a partition. The case

k
=
2

k = 2

is of special interest since the number

f
(
n
)

f(n)

of semisimple rings with n elements when

n
=

q
1

l
1

q
2

l
2

…

n = q_1^{{l_1}}q_2^{{l_2}} \ldots

is given by

f
(
n
)
=

P
2

(

l
1

)

P
2

(

l
2

)
…

f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots

.