bluuebunny/crossref_metadata_2025_split huggingface dataset:
Asymptotic behavior of linear integrodifferential systems
Authors: Barbu Viorel , Grossman Stanley I. 
Published: July 2010

We consider the system

(

L)

y
′

(
t
)
=
A
y
(
t
)
+

∫

−
∞

t

B
(
t
−
s
)
y
(
s
)
d
s
,
y
(
t
)
=
f
(
t
)
,
t
⩽
0

({\text {L)}}y’(t) = Ay(t) + \int _{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0}

where

y
(
t
)

y(t)

is an

n
n

-vector and

A
A

and

B
(
t
)

B(t)

are

n
×
n

n \times n

matrices. System

(

L)

({\text {L)}}

generates a semigroup given by

T
t

f
(
s
)
=
y
(
t
+
s
;
f
)

{T_t}f(s) = y(t + s;f)

for

f
f

bounded, continuous and having a finite limit at

−
∞

- \infty

. Under hypotheses concerning the roots of

det
(
λ
I
−
A
−

B
^

(
λ
)
)

\det (\lambda I - A - \hat B(\lambda ))

, where

B
^

(
λ
)

\hat B(\lambda )

is the Laplace transform, various results about the asymptotic behavior of

y
(
t
)

y(t)

are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If

B
(
t
)
∈

L
1

[
0
,
∞
)

B(t) \in {L^1}[0,\infty )

and

(
λ
I
−
A
−

B
^

(
λ
)

)

−
1

{(\lambda I - A - \hat B(\lambda ))^{ - 1}}

exists for

Re
⁡
λ
>
0

\operatorname {Re} \lambda > 0

, then for every

ϵ
>
0

\epsilon > 0

, there is an

M

ϵ

{M_{\epsilon }}

such that

|

|

T
t

f

|

|

⩽

M

ϵ

e

ϵ
t

|

|

f

|

|

||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f||

. Theorem 2. If

(
λ
I
−
A
−

B
^

(
λ
)

)

−
1

{(\lambda I - A - \hat B(\lambda ))^{ - 1}}

exists for

Re
⁡
λ
>
−
α
(
α
>
0
)

\operatorname {Re} \lambda > - \alpha (\alpha > 0)

and if

B
(
t
)

e

α
t

∈

L
1

[
0
,
∞
)

B(t){e^{\alpha t}} \in {L^1}[0,\infty )

, then the solution to

(

L)

({\text {L)}}

is exponentially asymptotically stable.
Automorphisms of a free associative algebra of rank 2. II
Authors: Czerniakiewicz Anastasia J. 
Published: July 2010

Let

R
R

be a commutative domain with 1.

R
⟨
x
,
y
⟩

R\langle x,y\rangle

stands for the free associative algebra of rank 2 over

R
;
R
[

x
~

,

y
~

]

R;R[\tilde x,\tilde y]

is the polynomial algebra over

R
R

in the commuting indeterminates

x
~

\tilde x

and

y
~

\tilde y

. We prove that the map

Ab
:
Aut
⁡
(
R
⟨
x
,
y
⟩
)
→
Aut
⁡
(
R
[

x
~

,

y
~

]
)

\text {Ab}: \operatorname {Aut} (R\langle x,y\rangle ) \to \operatorname {Aut} (R[\tilde x,\tilde y])

induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [5], Nagata [7] and van der Kulk [8]* that describes the automorphisms of

F
[

x
~

,

y
~

]

F[\tilde x,\tilde y]

(

F
F

a field) we are able to conclude that every automorphism of

F
⟨
x
,
y
⟩

F\langle x,y\rangle

is tame (i.e. a product of elementary automorphisms).
On bounded oscillation and asymptotic expansion of conformal strip mappings
Authors: Obrock Arthur E. 
Published: July 2010

Relations between the boundary parameters

ϕ
−

,

ϕ
+

{\phi _ - },{\phi _ + }

of a strip

S
=
{

ϕ
−

(
x
)
>
y
>

ϕ
+

(
x
)
}

S = \{ {\phi _ - }(x) > y > {\phi _ + }(x)\}

and the values

f
(
x
)

f(x)

of its canonical conformal mapping onto a horizontal strip

H
=
{

|

υ

|

>
1
}

H = \{ |\upsilon | > 1\}

are studied. Bounded oscillation

(

max
y

Re
⁡
f
(
x
+
i
y
)
−

min
y

Re
⁡
f
(
x
+
i
y
)
=
ω
(
x
)
=
O
(
1
)
)

({\max _y}\operatorname {Re} f(x + iy) - {\min _y}\operatorname {Re} f(x + iy) = \omega (x) = O(1))

is characterized in terms of

ϕ
−

,

ϕ
+

{\phi _ - },{\phi _ + }

. A formal series expansion

υ
=
∑

y
m

a

m
,
n

(
x
)

\upsilon = \sum {y^m}{a_{m,n}}(x)

is derived for the solution to the Dirichlet problem on

S
S

and its partial sums are used to obtain formulas for the asymptotic expansion of

f
f

in terms of

ϕ
+

,

ϕ
−

{\phi _ + },{\phi _ - }

.
Mean Cesàro summability of Laguerre and Hermite series
Authors: Poiani Eileen L. 
Published: July 2010

The primary purpose of this paper is to prove inequalities of the type

|

|

σ
n

(
f
,
x
)
W
(
x
)

|

|

p

⩽
C

|

|

f
(
x
)
W
(
x
)

|

|

p

||{\sigma _n}(f,x)W(x)|{|_p} \leqslant C||f(x)W(x)|{|_p}

where

σ
n

(
f
,
x
)

{\sigma _n}(f,x)

is the

n
n

th

(
C
,
1
)

(C,1)

mean of the Laguerre or Hermite series of

f
,
W
(
x
)

f, W(x)

is a suitable weight function of particular form,

C
C

is a constant independent of

f
(
x
)

f(x)

and

n
n

, and the norm is taken over

(
0
,
∞
)

(0,\infty )

in the Laguerre case and

(
−
∞
,
∞
)

( - \infty ,\infty )

in the Hermite case for

1
⩽
p
⩽
∞

1 \leqslant p \leqslant \infty

. Both necessary and sufficient conditions for these inequalities to remain valid are determined. For

p
>
∞

p > \infty

and

f
(
x
)
W
(
x
)
∈

L
p

f(x)W(x) \in {L^p}

, mean summability results showing that

lim

n
→
∞

|

|

[

σ
n

(
f
,
x
)
−
f
(
x
)
]
W
(
x
)

|

|

p

=
0

\lim \nolimits _{n \to \infty } ||[{\sigma _n}(f,x) - f(x)]W(x)|{|_p} = 0

are derived by use of the appropriate density theorems. Detailed proofs are presented for the Laguerre expansions, and the analogous results for Hermite series follow as corollaries.
Banach space bifurcation theory
Authors: Westreich David 
Published: July 2010

We consider the bifurcation problem for the nonlinear operator equation

x
=
λ
L
x
+
T
(
λ
,
x
,
y
)

x = \lambda Lx + T(\lambda ,x,y)

in a real Banach space

X
X

. Here

λ
0

{\lambda _0}

is an eigenvalue of the bounded linear operator

L
,
X
=
N
(
I
−

λ
0

L
)
⊕
R
(
I
−

λ
0

L
)
,
T
∈

C
1

L,X = N(I - {\lambda _0}L) \oplus R(I - {\lambda _0}L),T \in {C^1}

and

T
T

is of higher order in

x
x

. New techniques are developed to simplify the solution of the bifurcation problem. When

λ
0

{\lambda _0}

is a simple eigenvalue,

λ
0

{\lambda _0}

is shown to be a bifurcation point of the homogeneous equation (i.e.

y
≡
0

y \equiv 0

) with respect to 0. All solutions near

(

λ
0

,
0
)

({\lambda _0},0)

are shown to be of the form

(
λ
(
ϵ
)
,
x
(
ϵ
)
)
,
0
⩽

|

ϵ

|

>

ϵ
0

,
λ
(
ϵ
)

(\lambda (\epsilon ),x(\epsilon )),0 \leqslant |\epsilon | > {\epsilon _0},\lambda (\epsilon )

and

x
(
ϵ
)

x(\epsilon )

are continuous and

λ
(
ϵ
)

\lambda (\epsilon )

and

x
(
ϵ
)

x(\epsilon )

are in

C
n

{C^n}

or real analytic as

T
T

is in

C

n
+
1

{C^{n + 1}}

or is real analytic. When

T
T

is real analytic and

λ
(
ϵ
)

λ
0

\lambda (\epsilon ){\lambda _0}

then there are at most two solution branches, and each branch is an analytic function of

λ
\lambda

for

λ
≠

λ
0

\lambda \ne {\lambda _0}

. If

T
T

is odd and analytic, for each

λ
∈
(

λ
0

−
δ
,

λ
0

)

\lambda \in ({\lambda _0} - \delta ,{\lambda _0})

(or

λ
∈
(

λ
0

,

λ
0

+
δ
)

\lambda \in ({\lambda _0},{\lambda _0} + \delta )

) there exist two nontrivial solutions near 0 and there are no solutions near 0 for

λ
∈
(

λ
0

,

λ
0

+
δ
)

\lambda \in ({\lambda _0},{\lambda _0} + \delta )

(or

λ
∈
(

λ
0

−
δ
,

λ
0

)

\lambda \in ({\lambda _0} - \delta ,{\lambda _0})

). We then demonstrate that in each sufficiently small neighborhood of a solution of the homogeneous bifurcation problem there are solutions of the nonhomogeneous equation (i.e.

y
≢
0

y \not \equiv 0

) depending continuously on a real parameter and on

y
y

. If

λ
0

{\lambda _0}

is an eigenvalue of odd multiplicity we prove it is a point of bifurcation of the homogeneous equation. With a strong restriction on the projection of

T
T

onto the null space of

I
−

λ
0

L

I - {\lambda _0}L

we show

λ
0

{\lambda _0}

is a bifurcation point of the homogeneous equation when

λ
0

{\lambda _0}

is a double eigenvalue. Counterexamples to some of our results are given when the hypotheses are weakened.
Sets of formulas valid in finite structures
Authors: Selman Alan L. 
Published: July 2010

A function

V

\mathcal {V}

is defined on the set of all subsets of

ω
\omega

so that for each set K, the value,

V

K

{\mathcal {V}_K}

, is the set of formulas valid in all structures of cardinality in K. An analysis is made of the dependence of

V

K

{\mathcal {V}_K}

on K, For any set K, let

d

(
K
)

{\text {d}}(K)

be the Kleene-Post degree to which K belongs. It is easily seen that for all infinite sets K,

d

(
K
)
∨
1
≤
d
(

V

K

)
≤

d

(
K

)
′

{\text {d}}(K) \vee 1 \leq d({\mathcal {V}_K}) \leq {\text {d}}(K)’

. On the other hand, we prove that

d

(

V

K
∨
J

)
=

d

(

V

K

)
∨

d

(

V

J

)

{\text {d}}({\mathcal {V}_{K \vee J}}) = {\text {d}}({\mathcal {V}_K}) \vee {\text {d}}({\mathcal {V}_J})

, and use this to prove that, for any two degrees a and b,

a

≥
1
,

a

≤

b

≤

a

′

{\text {a}} \geq 1,{\text {a}} \leq {\text {b}} \leq {\text {a}}’

, and b r.e. a, there exists a set K so that

d

(
K
)
=

a

{\text {d}}(K) = {\text {a}}

and

d

(

V

K

)
=
b

{\text {d}}({\mathcal {V}_K}) = b

. Various similar results are also included.
A law of iterated logarithm for stationary Gaussian processes
Authors: Pathak Pramod K. , Qualls Clifford 
Published: July 2010

In this article the following results are established. Theorem A. Let

{
X
(
t
)
:
0
⩽
t
>
∞
}

\{ X(t):0 \leqslant t > \infty \}

be a stationary Gaussian process with continuous sample functions and

E
[
X
(
t
)
]
≡
0

E[X(t)] \equiv 0

. Suppose that the covariance function

r
(
t
)

r(t)

satisfies the following conditions. (a)

r
(
t
)
=
1
−

|

t

|

α

H
(
t
)
+
o
(

|

t

|

α

H
(
t
)
)

r(t) = 1 - |t{|^\alpha }H(t) + o(|t{|^\alpha }H(t))

as

t
→
0

t \to 0

, where

0
>
α
⩽
2

0 > \alpha \leqslant 2

and

H
H

varies slowly at zero, and (b)

r
(
t
)
=
O
(
1

/

log
⁡
t
)

r(t) = O(1/\log t)

as

t
→
∞

t \to \infty

Then for any nondecreasing positive function

ϕ
(
t
)

\phi (t)

defined on

[
a
,
∞
)

[a,\infty )

with

ϕ
(
∞
)
=
∞
,
P
[
X
(
t
)
>
ϕ
(
t
)

\phi (\infty ) = \infty ,P[X(t) > \phi (t)

i.o. for some sequence

t
n

→
∞
]
=
0
o
r
1

{t_n} \to \infty ] = 0or1

according as the integral

I
(
ϕ
)
=

∫
a
∞

g
(
ϕ
(
t
)
)
ϕ

(
t
)

−
1

exp
⁡
(
−

ϕ
2

(
t
)

/

2
)
d
t

I(\phi ) = \int _a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt}

is finite or infinite, where

g
(
x
)
=
1

/

σ

∼
−
1

(
1

/

x
)

g(x) = 1/_\sigma ^{ \sim - 1}(1/x)

is a regularly varying function with exponent

2

/

α

2/\alpha

and

σ

∼
2

(
t
)
=
2

|

t

|

α

H
(
t
)

_\sigma ^{ \sim 2}(t) = 2|t{|^\alpha }H(t)

.

Theorem C. Let

{

X
n

:
n
⩾
1
}

\{ {X_n}:n \geqslant 1\}

be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some

γ
>
0
,
r
(
n
)
=
O
(
1

/

n
γ

)

a
s

n
→
∞

\gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty

. Let

{
ϕ
(
n
)
:
n
⩾
1
}

\{ \phi (n):n \geqslant 1\}

be a nondecreasing sequence of positive numbers with

lim

n
→
∞

ϕ
(
n
)
=
∞

{\lim _{n \to \infty }}\phi (n) = \infty

; suppose that

Σ
(
1

/

ϕ
(
n
)
)
exp
⁡
(
−

ϕ
2

(
n
)

/

2
)
=
∞

\Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty

. Then
\[

lim

n
→
∞

∑

1
≤
k
≤
n

I
k

/

∑

1
≤
k
≤
n

E
[

I
k

]
=
1

a
.
s
.

,

\lim _{n \to \infty } \sum _{1 \leq k \leq n} I_k / \sum _{1 \leq k \leq n} E[I_k] = 1\quad \mathrm {a.s.},

\]
where

I
k

I_k

denotes the indicator function of the event

{

X
k

>
ϕ
(
k
)
}

\{ {X_k} > \phi (k)\}

.
Cobordism invariants, the Kervaire invariant and fixed point free involutions
Authors: Browder William 
Published: July 2010

Conditions are found which allow one to define an absolute version of the Kervaire invariant in

Z
2

{Z_2}

of a

Wu -

(
q
+
1
)

{\text {Wu - }}(q + 1)

oriented 2q-manifold. The condition is given in terms of a new invariant called the spectral cobordism invariant. Calculations are then made for the Kervaire invariant of the n-fold disjoint union of a manifold M with itself, which are then applied with

M
=

P

2
q

M = {P^{2q}}

, the real protective space. These give examples where the Kervaire invariant is not defined, and other examples where it has value

1
∈

Z

2

1 \in {{\mathbf {Z}}_2}

. These results are then applied to construct examples of smooth fixed point free involutions of homotopy spheres of dimension

4
k
+
1

4k + 1

with nonzero desuspension obstruction, of which some Brieskorn spheres are examples (results obtained also by Berstein and Giffen). The spectral cobordism invariant is also applied directly to these examples to give another proof of a result of Atiyah-Bott. The question of which values can be realized as the sequence of Kervaire invariants of characteristic submanifolds of a smooth homotopy real projective space is discussed with some examples. Finally a condition is given which yields smooth embeddings of homotopy

P
m

{P^m}

’s in

R

m
+
k

{R^{m + k}}

(which has been applied by E. Rees).
Existence of solutions of ordinary differential equations with generalized boundary conditions
Authors: Bernfeld Stephen R. , Lakshmikantham V. 
Published: July 2010

An investigation of the existence of solutions of the nonlinear boundary value problem

x
′

=
f
(
t
,
x
,
y
)
,

y
′

=
g
(
t
,
x
,
y
)
,
A
V
(
a
,
x
(
a
)
,
y
(
a
)
)
+
B
W
(
a
,
x
(
a
)
,
y
(
a
)
)
=

C
1

,
C
V
(
b
,
x
(
b
)
,
y
(
b
)
)
+
D
W
(
b
,
x
(
b
)
,
y
(
b
)
)
=

C
2

x’ = f(t,x,y),y’ = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}

, is made. Here we assume

g
,
f
:
[
a
,
b
]
×

R
p

×

R
q

→

R
p

g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}

are continuous, and

V
,
W
:
[
a
,
b
]
×

R
p

×

R
q

→
R

V,W:[a,b] \times {R^p} \times {R^q} \to R

are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.
Universal generators for varieties of nuclear spaces
Authors: Rosenberger B. 
Published: July 2010

It is shown that a product of several copies of

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

is a universal

ϕ
\phi

-nuclear space if the power series space

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

with

β
k
ϕ

=
−
log
⁡
(

ϕ

−
1

(
1

/

k
+
1

)
)
,
k

ϵ

{
0
,
1
,
2
,
⋯
}

\beta _k^\phi = - \log ({\phi ^{ - 1}}(1/\sqrt {k + 1} )),k\;\epsilon \;\{ 0,1,2, \cdots \}

, is

ϕ
\phi

-nuclear; here

ϕ
=
[
0
,
∞
)
→
[
0
,
∞
)

\phi = [0,\infty ) \to [0,\infty )

is a continuous, strictly increasing subadditive function with

ϕ
(
0
)
=
0

\phi (0) = 0

. In case

Λ
(

β
ϕ

)

\Lambda ({\beta ^\phi })

is not

ϕ
\phi

-nuclear the sequence space

Λ
(

l
ϕ
+

)

\Lambda (l_\phi ^ + )

is a

ϕ
\phi

-nuclear space with the property that every

ϕ
\phi

-nuclear space is isomorphic to a subspace of a product of

Λ
(

l
ϕ
+

)

\Lambda (l_\phi ^ + )

if

lim

sup

t
→
0

(
ϕ
(
t
)

)

−
1

ϕ
(

t

)
>
∞

{\lim \;\sup _{t \to 0}}{(\phi (t))^{ - 1}}\phi (\sqrt t ) > \infty

.
On Green’s function of an 𝑛-point boundary value problem
Authors: Das K. M. , Vatsala A. S. 
Published: July 2010

The Green’s function

g
n

(
x
,
s
)

{g_n}(x,s)

for an n-point boundary value problem,

y

(
n
)

(
x
)
=
0
,
y
(

a
1

)
=
y
(

a
2

)
=
⋯
=
y
(

a
n

)
=
0

{y^{(n)}}(x) = 0,y({a_1}) = y({a_2}) = \cdots = y({a_n}) = 0

is explicitly given. As a tool for discussing

sgn
⁡

g
n

(
x
,
s
)

\operatorname {sgn} g_n(x,s)

on the square

[

a
1

,

a
n

]
×
[

a
1

,

a
n

]

[{a_1},{a_n}] \times [{a_1},{a_n}]

, some results about polynomials with coefficients as symmetric functions of a’s are obtained. It is shown that
\[

∫

a
1

a
n

|

g
n

(
x
,
s
)

|

d
s

\int _{{a_1}}^{{a_n}} {|{g_n}(x,s)|ds}

\]
is a suitable polynomial in x. Applications to n-point boundary value problems and lower bounds for

a
m

(
m
≥
n
)

{a_m}\;(m \geq n)

are included.
A fixed point theorem, a perturbed differential equation, and a multivariable Volterra integral equation
Authors: Lovelady David Lowell 
Published: July 2010

A fixed point theorem is obtained for an equation of the form

u
=
T
[
p
,
f
+
G
[
u
]
]

u = T[p,f + G[u]]

. This theorem is then applied to a functionally perturbed ordinary differential equation of the form

u
′

(
t
)
=
f
(
t
)
+
A
(
t
,
u
(
t
)
)
+
G
[
u
]
(
t
)
;
u
(
0
)
=
p

u’(t) = f(t) + A(t,u(t)) + G[u](t);u(0) = p

, and, as a consequence of this, Fredholm integrodifferential equations of the form

z
′

(
t
)
=
f
(
t
)
+
ϕ
(
t
,
z
(
t
)
)
+

∫
0
∞

α
(
t
,
s
)
ω
(
z
(
s
)
)
d
s
,
z
(
0
)
=
p

z’(t) = f(t) + \phi (t,z(t)) + \int _0^\infty {\alpha (t,s)\omega (z(s))ds,z(0) = p}

. Applications are also made to a multivariable Volterra integral system
\[

u
1

(
s
,
t
)
=

g
1

(
s
,
t
)
+

∫
0
s

∫
0
t

F

[

u
1

(
x
,
y
)
,

u
2

(
x
,
y
)
,

u
3

(
x
,
y
)

]

d
y

d
x

u
2

(
s
,
t
)
=

g
2

(
s
,
t
)
+

∫
0
t

F

[

u
1

(
s
,
y
)
,

u
2

(
s
,
y
)
,

u
3

(
s
,
y
)

]

d
y

u
3

(
s
,
t
)
=

g
3

(
s
,
t
)
+

∫
0
s

F

[

u
1

(
x
,
t
)
,

u
2

(
x
,
t
)
,

u
3

(
x
,
t
)

]

d
x
,

\begin {array}{*{20}{c}} {{u_1}(s,t) = {g_1}(s,t) + \int _0^s {\int _0^t {F\left [ {{u_1}(x,y),{u_2}(x,y),{u_3}(x,y)} \right ]dy\;dx} } } \hfill \\ {{u_2}(s,t) = {g_2}(s,t) + \int _0^t {F\left [ {{u_1}(s,y),{u_2}(s,y),{u_3}(s,y)} \right ]} \;dy} \hfill \\ {{u_3}(s,t) = {g_3}(s,t) + \int _0^s {F\left [ {{u_1}(x,t),{u_2}(x,t),{u_3}(x,t)} \right ]} \;dx,} \hfill \\ \end {array}

\]
and, as a corollary to this, a differential equation of the form
\[

∂
2

∂
s
∂
t

u
(
s
,
t
)
=
f
(
s
,
t
)
+
F

[

u
(
s
,
t
)
,

∂

∂
s

u
(
s
,
t
)
,

∂

∂
t

u
(
s
,
t
)

]

,

u
(
s
,
0
)
=
ϕ
(
s
)
,
u
(
0
,
t
)
=
ψ
(
t
)
.

\begin {array}{*{20}{c}} \hfill {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t) = f(s,t) + F\left [ {u(s,t),\frac {\partial }{{\partial s}}u(s,t),\frac {\partial }{{\partial t}}u(s,t)} \right ],} \\ \hfill {u(s,0) = \phi (s),u(0,t) = \psi (t).} \\ \end {array}

\]
These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as
\[

∂
2

∂
s
∂
t

u
(
s
,
t
,
z
)
=
f
(
s
,
t
,
z
)
+
H

(

z
,
u
(
s
,
t
,
z
)
,

∂

∂
s

u
(
s
,
t
,
z
)
,

∂

∂
t

u
(
s
,
t
,
z
)

)

+

∫
0
1

K

(

z
,
r
,
u
(
s
,
t
,
r
)
,

∂

∂
s

u
(
s
,
t
,
r
)
,

∂

∂
t

u
(
s
,
t
,
r
)

)

d
r
,

u
(
s
,
0
,
z
)
=
σ
(
s
,
z
)
,
u
(
0
,
t
,
z
)
=
τ
(
t
,
z
)
.

\begin {array}{*{20}{c}} {\frac {{{\partial ^2}}}{{\partial s\partial t}}u(s,t,z) = f(s,t,z) + H\left ( {z,u(s,t,z),\frac {\partial }{{\partial s}}u(s,t,z),\frac {\partial }{{\partial t}}u(s,t,z)} \right )} \\ { + \int _0^1 {K\left ( {z,r,u(s,t,r),\frac {\partial }{{\partial s}}u(s,t,r),\frac {\partial }{{\partial t}}u(s,t,r)} \right )} \;dr,} \\ {u(s,0,z) = \sigma (s,z),u(0,t,z) = \tau (t,z).} \\ \end {array}

\]
Maximal regular right ideal space of a primitive ring. II
Authors: Koh Kwangil , Luh Hang 
Published: July 2010

If

R
R

is a ring, let

X
(
R
)

X(R)

be the set of maximal regular right ideals of

R
R

. For each nonempty subset

E
E

of

R
R

, define the hull of

E
E

to be the set

{
I
ϵ

X
(
R
)

|

E
⊆
I
}

\{ I \epsilon \, X(R)|\ E \subseteq I\}

and the support of

E
E

to be the complement of the hull of

E
E

. Topologize

X
(
R
)

X(R)

by taking the supports of right ideals of

R
R

as a subbase. If

R
R

is a right primitive ring, then

X
(
R
)

X(R)

is homeomorphic to an open subset of a compact space

X
(

R
#

)

X({R^\# })

of a right primitive ring

R
#

{R^\# }

, and

X
(
R
)

X(R)

is a discrete space if and only if

X
(
R
)

X(R)

is a compact Hausdorff space if and only if either

R
R

is a finite ring or a division ring. Call a closed subset

F
F

of

X
(
R
)

X(R)

a line if

F
F

is the hull of

I
∩
J

I \cap J

for some two distinct elements

I
I

and

J
J

in

X
(
R
)

X(R)

. If

R
R

is a semisimple ring, then every line contains an infinite number of points if and only if either

R
R

is a division ring or

R
R

is a dense ring of linear transformations of a vector space of dimension two or more over an infinite division ring such that every pair of simple (right)

R
R

-modules are isomorphic.
Existence of sum and product integrals
Authors: Helton Jon C. 
Published: July 2010

Functions are from

R
×
R

R \times R

to R, where R represents the set of real numbers. If c is a number and either (1)

∫
a
b

G
2

\smallint _a^b{G^2}

exists and

∫
a
b

G

\smallint _a^bG

exists, (2)

∫
a
b

G

\smallint _a^bG

exists and

a

Π

b

(
1
+
G
)

_a{{\mathbf {\Pi }}^b}(1 + G)

exists and is not zero or (3) each of

a

Π

b

(
1
+
G
)

_a{{\mathbf {\Pi }}^b}(1 + G)

and

a

Π
b

(
1
−
G
)

_a{\Pi ^b}(1 - G)

exists and is not zero, then

∫
a
b

c
G

\smallint _a^bcG

exists,

∫
a
b

|

c
G
−
∫
c
G

|

=
0

,
x

Π

y

(
1
+
c
G
)

\smallint _a^b|cG - \smallint cG| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + cG)

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and

∫
a
b

|

1
+
c
G
−

Π

(
1
+
c
G
)

|

=
0

\smallint _a^b|1 + cG - {\mathbf {\Pi }}(1 + cG)| = 0

. Furthermore, if H is a function such that

lim

x
→

p
−

H
(
x
,
p
)
,

lim

x
→

p
+

H
(
p
,
x
)
,

lim

x
,
y
→

p
−

H
(
x
,
y
)

{\lim _{x \to {p^ - }}}H(x,p),{\lim _{x \to {p^ + }}}H(p,x),{\lim _{x,y \to {p^ - }}}H(x,y)

and

lim

x
,
y
→

p
+

H
(
x
,
y
)

{\lim _{x,y \to {p^ + }}}H(x,y)

exist for each

p
∈
[
a
,
b
]
,
n
≥
2

p \in [a,b],n \geq 2

is an integer, and G satisfies either (1), (2) or (3) of the above, then

∫
a
b

H

G
n

\smallint _a^bH{G^n}

exists,

∫
a
b

|

H

G
n

−
∫
H

G
n

|

=
0

,
x

Π

y

(
1
+
H

G
n

)

\smallint _a^b|H{G^n} - \smallint H{G^n}| = 0{,_x}{{\mathbf {\Pi }}^y}(1 + H{G^n})

exists for

a
≤
x
>
y
≤
b

a \leq x > y \leq b

and

∫
a
b

|

1
+
H

G
n

−

Π

(
1
+
H

G
n

)

|

=
0

\smallint _a^b|1 + H{G^n} - {\mathbf {\Pi }}(1 + H{G^n})| = 0

.
Canonical forms and principal systems for general disconjugate equations
Authors: Trench William F. 
Published: July 2010

It is shown that the disconjugate equation (1)

L
x
≡
(
1

/

β
n

)
(
d

/

d
t
)
⋅
(
1

/

β

n
−
1

)
⋯
(
d

/

d
t
)
(
1

/

β
1

)
(
d

/

d
t
)
(
x

/

β
0

)
=
0

Lx \equiv (1/{\beta _n})(d/dt) \cdot (1/{\beta _{n - 1}}) \cdots (d/dt)(1/{\beta _1})(d/dt)(x/{\beta _0}) = 0

, a

>
t
>
b

> t > b

, where

β
i

>
0

{\beta _i} > 0

, and (2)

β
i

∈
C
(
a
,
b
)

{\beta _i} \in C(a,b)

, can be written in essentially unique canonical forms so that

∫
b

β
i

d
t
=
∞
(

∫
a

β
i

d
t
=
∞
)

{\smallint ^b}{\beta _i}dt = \infty ({\smallint _a}{\beta _i}dt = \infty )

for

1
≤
i
≤
n
−
1

1 \leq i \leq n - 1

. From this it follows easily that (1) has solutions

x
1

,
…
,

x
n

{x_1}, \ldots ,{x_n}

which are positive in (a, b) near

b
(
a
)

b(a)

and satisfy

lim

t
→
b

−

x
i

(
t
)

/

x
j

(
t
)
=
0
(

lim

t
→
a

+

x
i

(
t
)

/

x
j

(
t
)
=
∞
)

{\lim _{t \to b}} - {x_i}(t)/{x_j}(t) = 0({\lim _{t \to a}} + {x_i}(t)/{x_j}(t) = \infty )

for

1
≤
i
>
j
≤
n

1 \leq i > j \leq n

. Necessary and sufficient conditions are given for (1) to have solutions

y
1

,
…
,

y
n

{y_1}, \ldots ,{y_n}

such that

lim

t
→
b

−

y
i

(
t
)

/

y
j

(
t
)
=

lim

t
→
a

+

y
j

(
t
)

/

y
i

(
t
)
=
0

{\lim _{t \to b}} - {y_i}(t)/{y_j}(t) = {\lim _{t \to a}} + {y_j}(t)/{y_i}(t) = 0

for

1
≤
i
>
j
≤
n

1 \leq i > j \leq n

. Using different methods, P. Hartman, A. Yu. Levin and D. Willett have investigated the existence of fundamental systems for (1) with these properties under assumptions which imply the stronger condition

(

2
′

)

β
i

∈

C

(
n
−
i
)

(
a
,
b
)

(2’){\beta _i} \in {C^{(n - i)}}(a,b)

.