bluuebunny/crossref_metadata_2025_split huggingface dataset:
Polynomials related to the Bessel functions
Authors: Howard F. T. 
Published: July 2010

In this paper we examine the polynomials

W
n

(
a
)

{W_n}(a)

defined by means of
\[

−
4

e

x
a

[
x
(

e
x

−
1
)
−
2
(

e
x

+
1
)

]

−
1

=

∑

n
=
0

∞

W
n

(
a
)

x
n

/

n
!

.

- 4{e^{xa}}{[x({e^x} - 1) - 2({e^x} + 1)]^{ - 1}} = \sum \limits _{n = 0}^\infty {{W_n}(a){x^n}/n!} .

\]
These polynomials are closely related to the zeros of the Bessel function of the first kind of index —3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of

W
n

(
a
)

{W_n}(a)

are investigated. We show that if

n
n

is even then

W
n

(
a
)
>
0

{W_n}(a) > 0

for all

a
a

, and if

n
n

is odd then

W
n

(
a
)

{W_n}(a)

has only the one real root

a
=
1

/

2

a = 1/2

. Also we find upper and lower bounds for all

b
b

such that

W
n

(
a
+
b
i
)
=
0

{W_n}(a + bi) = 0

. The problem of multiple roots is discussed and we show that if

n
≡
0
,
1
,
5
,
8

n \equiv 0,1,5,8

or 9

(
mod

12
)

(\bmod \; 12)

, then

W
n

(
a
)

{W_n}(a)

has no multiple roots. Finally, if

n
≡
0
,
1
,
2
,
5
,
6

n \equiv 0,1,2,5,6

or 8

(
mod

12
)

(\bmod \; 12)

, then

W
n

(
a
)

{W_n}(a)

has no factor of the form

a
2

+
c
a
+
d

{a^2} + ca + d

where

c
c

and

(
mod

12
)

(\bmod \; 12)

are integers.
On bounded elements of linear algebraic groups
Authors: Sit Kwan-Yuk Claire 
Published: July 2010

Let

F
F

be a local field of characteristic zero and

G

{\text {G }}

a connected algebraic group defined over

F
F

. Let

G
G

be the locally compact group of

F
F

-rational points. One characterizes the group

B
(
G
)

B(G)

of

g
∈
G

g \in G

whose conjugacy class is relatively compact. For instance, if

G

{\text {G}}

is

F
F

-split or reductive without anisotropic factors then

B
(
G
)

B(G)

is the center of

G
G

. If

H
H

is a closed subgroup of

G
G

such that

G

/

H

G/H

has finite volume, then the centralizer of

H
H

in

G
G

is contained in

B
(
G
)

B(G)

. If, moreover,

H
H

is the centralizer of some

x
∈
G

x \in G

then

G

/

H

G/H

is compact.
Converses to the Ω-stability and invariant lamination theorems
Authors: Gottlieb Allan 
Published: July 2010

In 1967 Smale proved that for diffeomorphisms on closed smooth manifolds, Axiom

A

{\text {A}}

and no cycles are sufficient conditions for

Ω
\Omega

-stability and asserted the analogous theorem for vectorfields. Pugh and Shub have supplied a proof of the latter. Since then a major problem in dynamical systems has been Smale’s conjecture that Axiom

A

{\text {A}}

(resp.

A’

{\text {A’}}

) and no cycles are necessary as well as sufficient for

Ω
\Omega

-stability of diffeomorphisms (resp. vectorfields). Franks and Guckenheimer have worked on the diffeomorphism problem by strengthening the definition of

Ω
\Omega

-stable diffeomorphisms. In this paper it will be shown that an analogous strengthening of

Ω
\Omega

-stable vectorfields forces Smale’s conditions to be necessary. The major result of this paper is the following THEOREM. If

(
Λ
,
L
)

(\Lambda ,L)

is a compact laminated set,

N
N

is a normal bundle to the lamination, and

f
f

is an absolutely and differentiably

L
L

-stable diffeomorphism of a closed smooth manifold then

(

id -

f
#

¯

)
:

C
0

(
N
)
→

C
0

(
N
)

({\text {id - }}\overline {{f_\# }} ):{C^0}(N) \to {C^0}(N)

is surjective. If the lamination is just a compact submanifold, the theorem is already new. When applied to flows, this theorem gives the above result on

Ω
\Omega

-stable vectorfields.
The algebraic EHP sequence
Authors: Singer William M. 
Published: July 2010

Let

A
A

be the dual of the

mod
−
2

\bmod - 2

Steenrod algebra. If

M
,
N

M,N

, are graded unstable

A
A

-comodules, one can define and compute the derived functors

Coext
A

(
M
,
N
)

{\text {Coext} _A}(M,N)

using unstable injective resolutions of

N
N

. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact “EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the

E
,
H

E,H

, and

P
P

homomorphisms and the composition pairing

Coext
A

(
N
,
R
)
⊗

Coext
A

(
M
,
N
)
→

Coext
A

(
M
,
R
)

{\text {Coext} _A}(N,R) \otimes {\text {Coext} _A}(M,N) \to {\text {Coext} _A}(M,R)

. Among our results is a formula that measures the failure of the composition product to commute.
Immersions of complex hypersurfaces
Authors: Samsky Stanley R. 
Published: July 2010

The varieties

V
n

(
d
)
=
{
[

z
0

,
…
,

z
n

]
∈
C

P
n

:

z
0
d

+
⋯
+

z
n
d

=
0
,
d
>
0
}

{V^n}(d) = \{ [{z_0}, \ldots ,{z_n}] \in C{P^n}:z_0^d + \cdots + z_n^d = 0,d > 0\}

form a class of manifolds containing the complex projective spaces. Maps from

V
n

(
d
)

{V^n}(d)

to

V
k

(
e
)

{V^k}(e)

are partially characterized by a “degree". We prove some nonimmersion results which are phrased in terms of this degree, and which generalize the results of S. Feder [4] on complex projective spaces.
A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory
Authors: Zarrow Robert 
Published: July 2010

The (extended) symplectic modular group

(

∧
n

)

Γ
n

({ \wedge _n}){\Gamma _n}

is the set of all

2
n
×
2
n

2n \times 2n

integer matrices

M
M

such that
\[

(
M

J
t

M
=
±
J
)
,
M

J
t

M
=
J
,
J
=

[

0

a
m
p
;
I

−
I

a
m
p
;
0

]

,

(M{J^t}M = \pm J),M{J^t}M = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ { - I} & 0 \\ \end {array} } \right ],

\]

I
I

being the

n
×
n

n \times n

identity matrix. Let

S
n

=
{
M
∈

∧
n

−

Γ
n

|

M
=
−

t

M
}

{S_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = - {}^tM\}

and

T
n

=
{
M
∈

∧
n

−

Γ
n

|

M
=

t

M
}

{T_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = {}^tM\}

. We say

M
∼
N

M \sim N

if there exists

K
∈

Γ
n

K \in {\Gamma _n}

such that

M
=
K
N

t

K

M = KN{}^tK

. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.
Effective lower bounds for some linear forms
Authors: Cusick T. W. 
Published: July 2010

It is proved that if

1
,
α
,
β

1, \alpha ,\beta

are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number

d
⩾
2

d \geqslant 2

, for any integers

x
0

,

x
1

,

x
2

{x_0},{x_1},{x_2}

such that

|

norm

(

x
0

+
α

x
1

+
β

x
2

)

|

⩽
d

|{\text {norm}}({x_0} + \alpha {x_1} + \beta {x_2})| \leqslant d

, there exist effectively computable numbers

c
>
0

c > 0

and

k
>
0

k > 0

depending only on

α
\alpha

and

β
\beta

such that

|

x
1

x
2

|

(
log
⁡

|

x
1

x
2

|

)

k
log
⁡
d

|

x
0

+
α

x
1

+
β

x
2

|

>
c

|{x_1}{x_2}|{(\log |{x_1}{x_2}|)^{k\log d}}|{x_0} + \alpha {x_1} + \beta {x_2}| > c

holds whenever

x
1

x
2

≠
0

{x_1}{x_2} \ne 0

. It would be of much interest to remove the dependence on d in the exponent of

log
⁡

|

x
1

x
2

|

\log |{x_1}{x_2}|

, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth’s Theorem.
On Kolmogorov’s inequalities 𝑓̃_{𝑝}≤𝐶_{𝑝}, 𝑓₁, 0<𝑝<1
Authors: Davis Burgess 
Published: July 2010

Let

μ
\mu

be a signed measure on the unit circle A of the complex plane satisfying

|

μ

|

(
A
)
>
∞

|\mu |(A) > \infty

, where

|

μ

|

(
A
)

|\mu |(A)

is the total variation of

μ
\mu

, and let

μ
~

\tilde \mu

be the conjugate function of

μ
\mu

. A theorem of Kolmogorov states that for each real number p between 0 and 1 there is an absolute constant

C
p

{C_p}

such that

(

(
2
π

)

−
1

∫
0

2
π

|

μ
~

(

e

i
θ

)

|

p

d
θ

)

1

/

p

⩽

C
p

|

μ

|

(
A
)

{({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \mu ({e^{i\theta }}){|^p}d\theta )^{1/p}} \leqslant {C_p}|\mu |(A)

. Here it is shown that measures putting equal and opposite mass at points directly opposite from each other on the unit circle, and no mass any place else, are extremal for all of these inequalities, that is, if

ν
\nu

is one of these measures the number

(

(
2
π

)

−
1

∫
0

2
π

|

ν
~

(

e

i
θ

)

|

p

d
θ

)

1

/

p

/

|

ν

|

(
A
)

{({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \nu ({e^{i\theta }}){|^p}d\theta )^{1/p}}/|\nu |(A)

is the smallest possible value for

C
p

{C_p}

. These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.
Hilbert transforms associated with plane curves
Authors: Nagel Alexander , Wainger Stephen 
Published: July 2010

Let

(
t
,
γ
(
t
)
)

(t,\gamma (t))

be a plane curve. Set

H
γ

f
(
x
,
y
)
=
p.v.

∫
f
(
x
−
t
,
y
−
γ
(
t
)
)
d
t

/

t

{H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t

for

f
∈

C
0
∞

(

R
2

)

f \in C_0^\infty ({R^2})

. For a large class of curves, the authors prove

‖

H
γ

f

‖

p

⩽

A
p

‖
f
‖

p

,
5

/

3
>
p
>
5

/

2

{\left \| {{H_\gamma }f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p},5/3 > p > 5/2

. Various examples are given to show that some condition on the curve

(
t
,
γ
(
t
)
)

(t,\gamma (t))

is necessary.
Strong differentiability properties of Bessel potentials
Authors: Deignan Daniel J. , Ziemer William P. 
Published: July 2010

This paper is concerned with the “strong”

L
p

{L_p}

differentiability properties of Bessel potentials of order

α
>
0

\alpha > 0

of

L
p

{L_p}

functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial

P
x

(
y
)

{P_x}(y)

of degree

k
⩽
α

k \leqslant \alpha

such that
\[

lim

sup

diam

(
S
)
→
0

(

diam

S

)

−
k

{

|

S

|

−
1

∫

|

f
(
y
)
−

P
x

(
y
)

|

p

d
y

}

1

/

p

=
0

\lim \sup \limits _{{\text {diam}}(S) \to 0} \;{({\text {diam}}\;S)^{ - k}}{\left \{ {|S{|^{ - 1}}\int {|f(y) - {P_x}(y){|^p}dy} } \right \}^{1/p}} = 0

\]
where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.
A resolvent for an iteration method for nonlinear partial differential equations
Authors: Neuberger J. W. 
Published: July 2010

For each of m and n a positive integer denote by

S
(
m
,
i
)

S(m,i)

the space of all real-valued symmetric i-linear functions on

E
m

,
i
=
1
,
2
,
…
,
n

{E_m},i = 1,2, \ldots ,n

. Denote by L a nonzero linear functional on

S
(
m
,
n
)

S(m,n)

, denote by f a real-valued analytic function on

E
m

×
R
×
S
(
m
,
1
)
×
⋯
×
S
(
m
,
[
n

/

2
]
)

{E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])

and denote by

α
\alpha

a member of

D
(
f
)

D(f)

. Denote by H the space of all real-valued functions U, analytic at the origin of

E
m

{E_m}

, so that

α
=
(
0
,
U
(
0
)
,

U
′

(
0
)
,
…
,

U

(
[
n

/

2
]
)

(
0
)
)

\alpha = (0,U(0),U’(0), \ldots ,{U^{([n/2])}}(0))

. For

U
∈
H
,

f
U

(
x
)
≡
f
(
x
,
U
(
x
)
,

U
′

(
x
)
,
…
,

U

(
[
n

/

2
]
)

(
x
)
)

U \in H,{f_U}(x) \equiv f(x,U(x),U’(x), \ldots ,{U^{([n/2])}}(x))

for all x for which this is defined. A one-parameter semigroup (nonlinear if

f
≠
0

f \ne 0

) K on H is constructed so that if

U
∈
K

U \in K

, then

K
(
λ
)
U

K(\lambda )U

converges, as

λ
→
∞

\lambda \to \infty

, to a solution Y to the partial differential equation

L

Y

(
n
)

=

f
Y

L{Y^{(n)}} = {f_Y}

. A resolvent j for this semigroup is determined so that

J
(
λ
)
U

J(\lambda )U

also converges to y as

λ
→
∞

\lambda \to \infty

and so that

J

(
λ

/

n

)
n

U

J{(\lambda /n)^n}U

converges to

K
(
λ
)
U

K(\lambda )U

as

n
→
∞

n \to \infty

. The solutions

Y
∈
H

Y \in H

of

L

Y

(
n
)

=

f
Y

L{Y^{(n)}} = {f_Y}

are precisely the fixed points of the semigroup K.
Second-order differential equations with fractional transition points
Authors: Olver F. W. J. 
Published: July 2010

An investigation is made of the differential equation
\[

d
2

w

/

d

x
2

=
{

u
2

(
x
−

x
0

)
λ

f
(
u
,
x
)
+
g
(
u
,
x
)

/

(
x
−

x
0

)
2

}
w
,

{d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,

\]
in which u is a large real (or complex) parameter,

λ
\lambda

is a real constant such that

λ
>
−
2

\lambda > -2

, x is a real (or complex) variable, and

f
(
u
,
x
)

f(u,x)

and

g
(
u
,
x
)

g(u,x)

are continuous (or analytic) functions of x in a real interval (or complex domain) containing

x
0

{x_0}

. The interval (or domain) need not be bounded. Previous results of Langer and Riekstins giving approximate solutions in terms of Bessel functions of order

1

/

(
λ
+
2
)

1/(\lambda + 2)

are extended and error bounds supplied.
Disjoint circles: a classification
Authors: Ebert Gary L. 
Published: July 2010

For q a prime-power, let

IP

(
q
)

{\text {IP}}(q)

denote the miquelian inversive plane of order q. The classification of certain translation planes of order

q
2

{q^2}

, called subregular, has been reduced to the classification of sets of disjoint circles in

IP

(
q
)

{\text {IP}}(q)

. While R. H. Bruck has extensively studied triples of disjoint circles, this paper is concerned with sets of four or more circles in

IP

(
q
)

{\text {IP}}(q)

. In a previous paper, the author has shown (for odd q) that the number of quadruples of disjoint circles in

IP

(
q
)

{\text {IP}}(q)

is asymptotic to

q

12

/

1536

{q^{12}}/1536

. Hence a judicious approach to the classification problem is to study “interesting” quadruples. In general, let

C
1

,
…
,

C
n

{C_1}, \ldots ,{C_n}

be a nonlinear set of n disjoint circles in

IP

(
q
)

{\text {IP}}(q)

. Let H be the subgroup of the collineation group of

IP

(
q
)

{\text {IP}}(q)

composed of collineations that permute the

C
i

{C_i}

’s among themselves, and let K be that subgroup composed of collineations fixing each of the

C
i

′

s

{C_i}’s

. An interesting set of n disjoint circles would be one for which

K
=
1

K = 1

. It is shown that

K
=
1

K = 1

if and only if

(

∗

)

{

(i)

there

does

not

exist

a

circle

D

orthogonal

to

each

of

the

given

n

circles,

and

(ii)

we

do

not

have

one

circle

in

our

set

orthogonal

to

each

of

the

other

n
−
1

circles

.

\begin{equation}\tag {$*$} \left \{ {\begin {array}{*{20}{c}} {{\text {(i)}}\;{\text {there}}\;{\text {does}}\;{\text {not}}\;{\text {exist}}\;{\text {a}}\;{\text {circle}}\;D\;{\text {orthogonal}}\;{\text {to}}} \\ {{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {given}}\;n\;{\text {circles,}}\;{\text {and}}} \\ {{\text {(ii)}}\;{\text {we}}\;{\text {do}}\;{\text {not}}\;{\text {have}}\;{\text {one}}\;{\text {circle}}\;{\text {in}}\;{\text {our}}\;{\text {set}}\;{\text {orthogonal}}} \\ {{\text {to}}\;{\text {each}}\;{\text {of}}\;{\text {the}}\;{\text {other}}\;n - 1\;{\text {circles}}{\text {.}}} \\ \end{array} } \right .\end{equation}

When

n
=
4

n = 4

and under mild restrictions on q, an algorithm is developed that finds all nonlinear quadruples of disjoint circles satisfying the orthogonality conditions

(
∗
)

( \ast )

and having nontrivial group H. Given such a quadruple, the algorithm determines exactly what group H is acting. It is also shown that most quadruples in

IP

(
q
)

{\text {IP}}(q)

, for large q, do indeed satisfy the conditions

(
∗
)

( \ast )

. In addition, the cases when

n
=
5
,
6
,

n = 5,6,

or 7 are explored to a lesser degree.
Distribution of eigenvalues in the presence of higher order turning points
Authors: Leung Anthony 
Published: July 2010

This article is concerned with the eigenvalue problem

u
(
x
)
−

λ
2

p
(
x
)
u
(
x
)
=
0
,
u
(
x
)
∈

L
2

(
−
∞
,
∞
)

u(x) - {\lambda ^2}p(x)u(x) = 0,u(x) \in {L_2}( - \infty ,\infty )

, where

p
(
x
)

p(x)

is real, analytic and possesses zeroes of arbitrary orders. Under certain conditions on

p
(
x
)

p(x)

, approximate formulas for the eigenvalues are found. The problem considered is of interest in the study of particle scattering and wave mechanics. The formula is analogous to the quantum rule of Bohr-Sommerfeld.
On the sequence space 𝑙_{(𝑝_{𝑛})} and 𝜆_{(𝑝_{𝑛})}, 0<𝑝_{𝑛}≤1
Authors: Schonefeld S. A. , Stiles W. J. 
Published: July 2010

Let

(

p
n

)

({p_n})

and

(

q
n

)

({q_n})

be sequences in the interval

(
0
,
1
]

(0,1]

, let

l

(

p
n

)

{l_{({p_n})}}

be the set of all real sequences

(

x
n

)

({x_n})

such that

∑

|

x
n

|

p
n

>
∞

\sum {|{x_n}{|^{{p_n}}} > \infty }

, and let

λ

(

q
n

)

{\lambda _{({q_n})}}

be the set of all real sequences

(

y
n

)

({y_n})

such that

sup
π

∑

|

y
n

|

q

π
(
n
)

>
∞

{\sup _\pi }\sum {|{y_n}{|^{{q_{\pi (n)}}}} > \infty }

where the sup is taken over all permutations

π
\pi

of the positive integers. The purpose of this paper is to investigate some of the properties of these spaces. Our results are primarily concerned with (1) conditions which are necessary and/or sufficient for

l

(

p
n

)

{l_{({p_n})}}

(resp.,

λ

(

p
n

)

{\lambda _{({p_n})}}

) to equal

l

(

q
n

)

{l_{({q_n})}}

(resp.,

λ

(

q
n

)

{\lambda _{({q_n})}}

), and (2) isomorphic and topological properties of the subspaces of these spaces. In connection with (1), we show that the following four conditions are equivalent for any sequence

(

ε
n

)

({\varepsilon _n})

which decreases to zero and has

ε
1

>
1

{\varepsilon _1} > 1

. (a) There exists a number

K
>
1

K > 1

such that the series

∑

1

/

K

1

/

ε
n

\sum {1/{K^{1/{\varepsilon _n}}}}

converges; (b) the elements

ε
n

{\varepsilon _n}

of the sequence satisfy the condition

ε
n

=
O
(
1

/

ln
⁡
n
)

{\varepsilon _n} = O(1/\ln n)

; (c) the sequence

(
(
ln
⁡
n
)
(
(
1

/

n
)

∑
1
n

ε
j

)
)

((\ln n)((1/n)\sum \nolimits _1^n {{\varepsilon _j}} ))

is bounded; and (d)

l

(
1
−

ε
n

)

{l_{(1 - {\varepsilon _n})}}

equals

l
1

{l_1}

. In connection with (2), we show that the following are true when

(

p
n

)

({p_n})

increases to one. (a)

λ

(

p
n

)

{\lambda _{({p_n})}}

contains an infinite-dimensional closed subspace where the

l

(

p
n

)

{l_{({p_n})}}

-topology and the

λ

(

p
n

)

{\lambda _{({p_n})}}

-topology agree; (b)

l

(

p
n

)

{l_{({p_n})}}

and

λ

(

p
n

)

{\lambda _{({p_n})}}

contain closed subspaces isomorphic to

l
1

{l_1}

; and (c)

λ

(

p
n

)

{\lambda _{({p_n})}}

contains no infinite-dimensional subspace where the

λ

(

p
n

)

{\lambda _{({p_n})}}

-topology agrees with the

l
1

{l_1}

-topology if and only if
\[

lim
(

(
1

/

n

)

p
1

+

(
1

/

n

)

p
2

+
⋯
+

(
1

/

n

)

p
n

)
=
∞
.

\lim ({(1/n)^{{p_1}}} + {(1/n)^{{p_2}}} + \cdots + {(1/n)^{{p_n}}}) = \infty .

\]
Bender groups as standard subgroups
Authors: Griess Robert L. , Mason David R. , Seitz Gary M. 
Published: July 2010

A subgroup X of a finite group G is called

∗

^ \ast

-standard if

X
~

=
X

/

O
(
X
)

\tilde X = X/O(X)

is quasisimple,

Y
=

C
G

(
X
)

Y = {C_G}(X)

is tightly embedded in G and

N
G

(
X
)
=

N
G

(
Y
)

{N_G}(X) = {N_G}(Y)

. This generalizes the notion of standard subgroups. Theorem. Let G be a finite group with

O
(
G
)
=
1

O(G) = 1

. Suppose X is

∗

^ \ast

-standard in G and

X
~

/

Z
(

X
~

)
≅

L
2

(

2
n

)
,

U
3

(

2
n

)

\tilde X/Z(\tilde X) \cong {L_2}({2^n}),{U_3}({2^n})

or

Sz

(

2
n

)

{\text {Sz}}({2^n})

. Assume

X
⋪
G

X \ntriangleleft G

. Then

O
(
X
)
=
1

O(X) = 1

and one of the following holds:

(

i

)

E
(
G
)
≅
X
×
X

({\text {i}})\;E(G) \cong X \times X

.

(

ii

)

X
≅

L
2

(

2
n

)

({\text {ii}})\;X \cong {L_2}({2^n})

and

E
(
G
)
≅

L
2

(

2

2
n

)
,

U
3

(

2
n

)

o
r

L
3

(

2
n

)

E(G) \cong {L_2}({2^{2n}}),{U_3}({2^n})\;or\;{L_3}({2^n})

.

(

iii

)

X
≅

U
3

(

2
n

)

({\text {iii}})\;X \cong {U_3}({2^n})

and

E
(
G
)
≅

L
3

(

2

2
n

)

E(G) \cong {L_3}({2^{2n}})

.

(

iv

)

X
≅

Sz

(

2
n

)

({\text {iv}})\;X \cong {\text {Sz}}({2^n})

and

E
(
G
)
≅

Sp

(
4
,

2
n

)

E(G) \cong {\text {Sp}}(4,{2^n})

.

(

v

)

X
≅

L
2

(
4
)

({\text {v}})\;X \cong {L_2}(4)

and

E
(
G
)
≅

M

12

,

A
9

,

J
1

,

J
2

,

A
7

,

L
2

(
25
)
,

L
3

(
5
)

o
r

U
3

(
5
)

E(G) \cong {M_{12}},{A_9},{J_1},{J_2},{A_7},{L_2}(25),{L_3}(5)\;or\;{U_3}(5)

.

(

vi

)

X
≅

Sz

(
8
)

({\text {vi}})\;X \cong {\text {Sz}}(8)

and

E
(
G
)
≅

Ru

E(G) \cong {\text {Ru}}

(the Rudvalis group).

(

vii

)

X
≅

L
2

(
8
)

({\text {vii}})\;X \cong {L_2}(8)

and

E
(
G
)
≅

G
2

(
3
)

E(G) \cong {G_2}(3)

.

(

viii

)

X
≅

SL

(
2
,
5
)

({\text {viii}})\;X \cong {\text {SL}}(2,5)

and G has sectional 2-rank at most 4. In particular, if G is simple,

G
≅

M

12

,

A
9

,

J
1

,

J
2

,

Ru

,

U
3

(
5
)
,

L
3

(
5
)
,

G
2

(
5
)
,
o
r

3

D
4

(
5
)

G \cong {M_{12}},{A_9},{J_1},{J_2},{\text {Ru}},{U_3}(5),{L_3}(5),{G_2}(5), or\;{^3}{D_4}(5)

.
A nonlinear semigroup for a functional differential equation
Authors: Brewer Dennis W. 
Published: July 2010

A representation theorem is obtained for solutions of the nonlinear functional differential equation

(

1

)

u
′

(
t
)
=
F
(

u
t

)
,
t
⩾
0
,

u
(
t
)
=
ϕ
(
t
)
,
t
⩽
0
,

\begin{equation}\tag {$1$} u’(t) = F({u_t}), t \geqslant 0,\quad u(t) = \phi (t), t \leqslant 0,\end{equation}

as a semigroup of nonlinear operators on a space of initial data X of “fading memory type.” Equation (1) is studied in the abstract setting of a Banach space E. The nonlinear functional F is a uniformly Lipschitz continuous mapping from X to E. The semigroup is constructed by transforming (1) to an abstract Cauchy problem

(

C
P

)

w
′

(
t
)
+
A
w
(
t
)
=
0
,

w
(
0
)
=
ϕ
,

\begin{equation}\tag {$CP$} w’(t) + Aw(t) = 0,\quad w(0) = \phi ,\end{equation}

in the space X and applying a generation theorem of M. Crandall and T. Liggett to the operator A in X. The case when (1) is a nonlinear Volterra integrodifferential equation of infinite delay is given special consideration. The semigroup representation is used to obtain finite difference approximations for solutions of (CP) and to study the continuity of solutions of (1) with respect to perturbations of F and

ϕ
\phi

.
The asymptotic behaviour of certain integral functions
Authors: Fenton P. C. 
Published: July 2010

Let

f
(
z
)

f(z)

be an integral function satisfying
\[

∫

∞

{
log

m
(
r
,
f
)

−

cos

π
ρ

log

M
(
r
,
f
)

}
+

d
r

r

ρ
+
1

>

∞

{\int _{}^\infty \{\log \,m(r,f)\, - \,\cos \,\pi \rho \,\log \,M(r,f)\} ^ + }\frac {{dr}}{{{r^{\rho + 1}}}}\, > \,\infty

\]
and
\[

0

>

lim

r

→
∞

¯

log

M
(
r
,
f
)

r
ρ

>

∞

0\, > \,\lim \limits _{\overline {r\, \to \infty } } \,\frac {{\log \,M(r,f)}}{{{r^\rho }}}\, > \,\infty

\]
for some

ρ
:

0

>

ρ

>

1

\rho :\,0\, > \,\rho \, > \,1

. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres

ζ
i

{\zeta _i}

and radii

t
i

{t_i}

for which
\[

∑

i
=
1

∞

t
i

|

ζ
i

|

>
∞

\sum \limits _{i = 1}^\infty {\frac {{{t_i}}}{{\left | {{\zeta _i}} \right |}}} > \infty

\]
.