bluuebunny/crossref_metadata_2025 huggingface dataset:
𝜔-morasses, and a weak form of Martin’s axiom provable in 𝑍𝐹𝐶
Authors: Velleman Dan
Published: June 2010

We prove, in

ZFC
\text {ZFC}

, that simplified gap-

1
1

morasses of height

ω
\omega

exist. By earlier work on the relationship between morasses and forcing it immediately follows that a certain Martin’s axiom-type forcing axiom is provable in

ZFC
\text {ZFC}

. We show that this forcing axiom can be thought of as a weak form of

MA

ω
1

{\text {MA}}_{\omega _1}

and give some applications.
On Fourier multiplier transformations of Banach-valued functions
Authors: McConnell Terry R.
Published: June 2010

We obtain analogues of the Mihlin multiplier theorem and Littlewood-Paley inequalities for functions with values in a suitable Banach space

B
B

. The requirement on

B
B

is that it have the unconditionality property for martingale difference sequences.
Real zeros of derivatives of meromorphic functions and solutions of second order differential equations
Authors: Hellerstein Simon , Shen Li-Chien , Williamson Jack
Published: June 2010

We classify all functions

F
F

meromorphic in the plane with only real zeros and real poles which satisfy the additional conditions that

F
′

F’

has no zeros and

F
F

only real zeros. We apply this classification, in combination with some earlier results, to the study of the reality of zeros of solutions of the equation

w
+
H
(
z
)
w
=
0
,
H

w + H(z)w = 0,H

entire.
A quasilinear hyperbolic integro-differential equation related to a nonlinear string
Authors: Heard Melvin L.
Published: June 2010

We discuss global existence, boundedness and regularity of solutions to the integrodifferential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row u Subscript t t Baseline left-parenthesis t comma x right-parenthesis plus c left-parenthesis t right-parenthesis u Subscript t Baseline left-parenthesis t comma x right-parenthesis minus upper M left-parenthesis integral Subscript negative normal infinity Superscript plus normal infinity Baseline StartAbsoluteValue u Subscript x Baseline left-parenthesis t comma s right-parenthesis EndAbsoluteValue squared d s right-parenthesis u Subscript x x Baseline left-parenthesis t comma x right-parenthesis plus u left-parenthesis t comma x right-parenthesis 2nd Row equals h left-parenthesis t comma x comma u left-parenthesis t comma x right-parenthesis right-parenthesis comma 0 less-than-or-equal-to t greater-than normal infinity comma x element-of bold upper R comma 3rd Row u left-parenthesis 0 comma x right-parenthesis equals u 0 left-parenthesis x right-parenthesis comma u Subscript t Baseline left-parenthesis 0 comma x right-parenthesis equals u 1 left-parenthesis x right-parenthesis comma x element-of bold upper R period EndLayout">

u

t

t

(
t
,
x
)
+
c

(
t
)

u
t

(
t
,
x
)
−
M

(

∫

−
∞

+
∞

|

u
x

(
t
,
s
)

|

2

d
s

)

u

x

x

(
t
,
x
)
+
u

(
t
,
x
)

=
h

(
t
,
x
,
u

(
t
,
x
)
)
,

0
≤
t
>
∞
,
x
∈

R

,

u

(
0
,
x
)
=

u
0

(
x
)
,

u
t

(
0
,
x
)
=

u
1

(
x
)
,

x
∈

R

.

\begin {array}{*{20}{c}} {{u_{t\,t}}\,(t,x) + c\,(t)\,{u_t}(t,x) - M\;\left ( {\int _{ - \infty }^{ + \infty } {|{u_x}(t,s){|^2}\;ds} } \right )\;{u_{x\,x}}(t,x) + u\,(t,x)} \\ { = h\,(t,x,u\,(t,x)), \qquad 0 \leq t > \infty ,x \in {\mathbf {R}},} \\ {u\,(0,x) = {u_0}(x),\quad {u_t}(0,x) = {u_1}(x), \qquad x \in {\mathbf {R}}.} \\ \end {array}

\]
This type of equation occurs in the study of the nonlinear behavior of elastic strings. We show that if the initial data

u
0

(
x
)
,

u
1

(
x
)

{u_0}\,(x),{u_1}\,(x)

is small in a suitable sense, and if the damping coefficient

c

(
t
)

c\,(t)

grows sufficiently fast, then the above equation possesses a globally defined classical solution for forcing terms

h

(
t
,
x
,
u
)

h\,(t,x,u)

which are sublinear in

u
u

. In the nonlinearity we require that

M
∈

C
1

[
0
,
∞
)

M \in {C^1}\,[0,\infty )

and, in addition, satisfies

M
(
λ
)
≥

m
0

>
0

M( \lambda ) \geq {m_0} > 0

for all

λ
≥
0

\lambda \geq 0

.
Generic representations are induced from square-integrable representations
Authors: Lipsman Ronald L.
Published: June 2010

It is proven for arbitrary real algebraic groups that the generic irreducible unitary representation is induced from a square-integrable representation (modulo the projective kernel). This generalizes the well-known result for reductive groups that the generic representations are either discrete series, or induced from discrete series (modulo the nilradical) representations of cuspidal parabolic subgroups.
𝑅-sets and category
Authors: Barua Rana
Published: June 2010

We prove some category theoretic results for

R
R

-sets much in the spirit of Vaught and Burgess. Since the proofs entail many results on

R
R

-sets and the

R
R

-operator, we have studied them in some detail and have formulated many results appropriate for our purpose in, perhaps, a more unified manner than is available in the literature. Our main theorem is the following: Any

R
R

-set in the product of two Polish spaces can be approximated, in category, uniformly over all sections, by sets generated by rectangles with one side an

R
R

-set and the other a Borel set. In fact, we prove a levelwise version of this result. For

C
C

-sets, this has been proved by V. V. Srivatsa.
The formation of the dead core in parabolic reaction-diffusion problems
Authors: Bandle Catherine , Stakgold Ivar
Published: June 2010

For some nonlinear parabolic problems of reaction-diffusion, a region of zero reactant concentration may be formed in finite time. Conditions are formulated for the existence of such a dead core and estimates for its time of onset are also given. These results complement previous ones that dealt with the stationary (elliptic) problem.
Triangulations of subanalytic sets and locally subanalytic manifolds
Authors: Shiota M. , Yokoi M.
Published: June 2010

If two polyhedrons are locally subanalytically homeomorphic (that is, the graph is locally subanalytic), they are

PL

{\text {PL}}

homeomorphic. A locally subanalytic manifold is one whose coordinate transformations are locally subanalytic. It is proved that a locally subanalytic manifold has a unique

PL

{\text {PL}}

manifold structure. A semialgebraic manifold also is considered.
Galois theory for cylindric algebras and its applications
Authors: Comer Stephen D.
Published: June 2010

A Galois correspondence between cylindric set algebras and permutation groups is presented in this paper. Moreover, the Galois connection is used to help establish two important algebraic properties for certain classes of finite-dimensional cylindric algebras, namely the amalgamation property and the property that epimorphisms are surjective.
Oriented manifolds that fiber over 𝑆⁴
Authors: Kahn Steven M.
Published: June 2010

Necessary and sufficient conditions are given for an oriented manifold

M
M

to fiber up to cobordism over the

4
4

-sphere

S
4

{S^4}

(i.e. for

M
M

to be oriented cobordant to a fiber bundle over

S
4

{S^4}

). The result here extends those previously obtained for fiberings over

S
1

{S^1}

and

S
2

{S^2}

. In addition, fiberings over products of surfaces are studied with complete solutions (in the sense above) being given in most cases including those of

S
2

×

S
2

{S^2} \times {S^2}

and

(

S
1

)
4

{({S^1})^4}

.
On invariant finitely additive measures for automorphism groups acting on tori
Authors: Dani S. G.
Published: June 2010

Consider the natural action of a subgroup

H
H

of

GL

(
n
,

Z

)

{\text {GL}}(n,{\mathbf {Z}})

on

T

n

{{\mathbf {T}}^n}

. We relate the

H
H

-invariant finitely additive measures on

(

T

n

,

L

)

({{\mathbf {T}}^n},\mathcal {L})

where

L

\mathcal {L}

is the class of all Lebesgue measurable sets, to invariant subtori

C
C

such that the

H
H

-action on either

C
C

or

T

n

/

C

{{\mathbf {T}}^n}/C

factors to an action of an amenable group. In particular, we conclude that if

H
H

is a nonamenable group acting irreducibly on

T

n

{{\mathbf {T}}^n}

then the normalised Haar measure is the only

H
H

-invariant finitely additive probability measure on

(

T

n

,

L

)

({{\mathbf {T}}^n},\mathcal {L})

such that

μ
(
R
)
=
0

\mu (R) = 0

, where

R
R

is the (countable) subgroup consisting of all elements of finite order; this answers a question raised by J. Rosenblatt. Along the way we analyse

H
H

-invariant finitely additive measures defined for all subsets of

T

n

{{\mathbf {T}}^n}

and deduce, in particular, that the Haar measure extends to an

H
H

-invariant finitely additive measure defined on all sets if and only if

H
H

is amenable.
A method for investigating geometric properties of support points and applications
Authors: Brown Johnny E.
Published: June 2010

A normalized univalent function

f
f

is a support point of

S
S

if there exists a continuous linear functional

L
L

(which is nonconstant on

S
S

) for which

f
f

maximizes

Re
⁡
L
(
g
)
,
g
∈
S

\operatorname {Re} L(g),g \in S

. For such functions it is known that

Γ
=

C

−
f
(
U
)

\Gamma = {\text {C}} - f(U)

is a single analytic arc that is part of a trajectory of a certain quadratic differential

Q
(
w
)

d

w
2

Q(w)\;d{w^2}

. A method is developed which is used to study geometric properties of support points. This method depends on consideration of

Im
⁡
{

w
2

Q
(
w
)
}

\operatorname {Im} \{ {w^2}Q(w)\}

rather than the usual

Re
⁡
{

w
2

Q
(
w
)
}

\operatorname {Re} \{ {w^2}Q(w)\}

. Qualitative, as well as quantitative, applications are obtained. Results related to the Bieberbach conjecture when the extremal functions have initial real coefficients are also obtained.
Fuchsian groups and algebraic number fields
Authors: Waterman P. L. , Maclachlan C.
Published: June 2010

Given the signature of a finitely-generated Fuchsian group, we find the minimal extension of the rationals for which there is a Fuchsian group having the required signature, whose matrix entries lie in this field.
The Dirac operator on spaces with conical singularities and positive scalar curvatures
Authors: Chou Arthur Weichung
Published: June 2010

We study, in the spirit of Jeff Cheeger, the Dirac operator on a space with conical singularities. We obtain a Bochner-type vanishing theorem and prove an index theorem in the singular case. Also, the relationship with manifolds with boundary is explored. In the Appendix two methods of deforming the metric near the boundary are established and applied to obtain several new results on constructing complete metrics with positive scalar curvature.
Regularity properties of solutions to the basic problem in the calculus of variations
Authors: Clarke F. H. , Vinter R. B.
Published: June 2010

This paper concerns the basic problem in the calculus of variations: minimize a functional

J
J

defined by
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J left-parenthesis x right-parenthesis equals integral Subscript a Superscript b Baseline upper L left-parenthesis t comma x left-parenthesis t right-parenthesis comma ModifyingAbove x With dot left-parenthesis t right-parenthesis right-parenthesis d t">

J
(
x
)
=

∫
a
b

L
(
t
,
x
(
t
)
,

x
˙

(
t
)
)

d
t

J(x) = \int _a^b {L(t,x(t),\dot x(t))\;dt}

\]
over a class of arcs

x
x

whose values at

a
a

and

b
b

have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution

x
x

, there exists a relatively open subset

Ω
\Omega

of

[
a
,
b
]

[a,b]

, of full measure, on which

x
x

is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when

L
L

is independent of

t
t

, the solution has essentially bounded derivative.
On the asphericity of ribbon disc complements
Authors: Howie James
Published: June 2010

The complement of a ribbon

n
n

-disc in the

(
n
+
2
)

(n + 2)

-ball has a

2
2

-dimensional spine which shares some of the combinatorial properties of classical knot complement spines. It is an open question whether such

2
2

-complexes are always aspherical. To any ribbon disc we associate a labelled oriented tree, from which the homotopy type of the complement can be recovered, and we prove asphericity in certain special cases described by conditions on this tree. Our main result is that the complement is aspherical whenever the associated tree has diameter at most

3
3

.
On the ideals of a Noetherian ring
Authors: Stafford J. T.
Published: June 2010

We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain

R
R

with a minimal, nonzero ideal

I
I

, such that

R

/

I

R/I

is a commutative polynomial ring in

n
n

variables, and a Noetherian domain

S
S

with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal

I
I

in a Noetherian domain

T
T

such that

T

/

I

T/I

is not Artinian.
Quotients by 𝐶*×𝐶* actions
Authors: Białynicki-Birula Andrzej , Sommese Andrew John
Published: June 2010

Let

T
≈

C

∗

×

C

∗

T \approx {{\mathbf {C}}^\ast } \times {{\mathbf {C}}^\ast }

act meromorphically on a compact Kähler manifold

X
X

, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if

X
X

is projective. PROBLEM. Classify all

T
T

-invariant open subsets

U
U

of

X
X

such that the geometric quotient

U
→
U

/

T

U \to U/T

exists with

U

/

T

U/T

a compact complex space (necessarily algebraic if

X
X

is). In this paper a simple to state and use solution to this problem is given. The classification of

U
U

is reduced to finite combinatorics. Associated to the

T
T

action on

X
X

is a certain finite

2
2

-complex

C

(
X
)

\mathcal {C}(X)

. Certain

{
0
,
1
}

\{ 0,1\}

valued functions, called moment measures, are defined in the set of

2
2

-cells of

C

(
X
)

\mathcal {C}(X)

. There is a natural one-to-one correspondence between the

U
U

with compact quotients,

U

/

T

U/T

, and the moment measures.
On differential equations associated with Euler product expressions
Authors: Knowles Ian
Published: June 2010

A method is given by which one may associate (uniquely) certain differential equations with analytic functions defined by certain Euler product expressions. Some of the consequences of this construction include results relating the location of the zeros of the analytic functions to asymptotic properties of the solutions of the differential equations, as well as a differential equation characterization of those Dirichlet series with multiplicative coefficients.
Six standard deviations suffice
Authors: Spencer Joel
Published: June 2010

Given

n
n

sets on

n
n

elements it is shown that there exists a two-coloring such that all sets have discrepancy at most

K

n

1

/

2

K{n^{1/2}}

,

K
K

an absolute constant. This improves the basic probabilistic method with which

K
=
c

(
ln
⁡
n

)

1

/

2

K = c{(\ln n)^{1/2}}

. The result is extended to

n
n

finite sets of arbitrary size. Probabilistic techniques are melded with the pigeonhole principle. An alternate proof of the existence of Rudin-Shapiro functions is given, showing that they are exponential in number. Given

n
n

linear forms in

n
n

variables with all coefficients in

[
−
1
,
+
1
]

[ - 1, + 1]

it is shown that initial values

p
1

,
…
,

p
n

∈
{
0
,
1
}

{p_1}, \ldots ,{p_n} \in \{ 0,1\}

may be approximated by

ε
1

,
…
,

ε
n

∈
{
0
,
1
}

{\varepsilon _1}, \ldots ,{\varepsilon _n} \in \{ 0,1\}

so that the forms have small error.
Characteristic classes of transversely homogeneous foliations
Authors: Benson Chal , Ellis David B.
Published: June 2010

The foliations studied in this paper have transverse geometry modeled on a homogeneous space

G

/

H

G/H

with transition functions given by the left action of

G
G

. It is shown that the characteristic classes for such a foliation are determined by invariants of a certain flat bundle. This is used to prove that when

G
G

is semisimple, the characteristic classes are rigid under smooth deformations, extending work of Brooks, Goldman and Heitsch.
The Euler equation for functionals with linear growth
Authors: Anzellotti Gabriele
Published: June 2010

We give a first variation formula for functionals of the type

∫
Ω

f
(
x
,
μ
)

\int _\Omega {f(x,\mu )}

, where

f
(
x
,
p
)
:
Ω
×

R

k

→

R

f(x,p):\Omega \times {{\mathbf {R}}^k} \to {\mathbf {R}}

is of linear growth in

p
p

for large

|

p

|

|p|

and

μ
\mu

is a

R

k

{{\mathbf {R}}^k}

-valued measure in

Ω
\Omega

. The Euler equation for the minima of various functionals defined on spaces of

BV

{\text {BV}}

functions is then studied.
Directed sets and cofinal types
Authors: Todorčević Stevo
Published: June 2010

We show that

1
,
ω
,

ω
1

,
ω
×

ω
1

1,\omega ,{\omega _1},\omega \times {\omega _1}

and

[

ω
1

]

>
ω

{[{\omega _1}]^{ > \omega }}

are the only cofinal types of directed sets of size

ℵ
1

{\aleph _1}

, but that there exist many cofinal types of directed sets of size continuum.
Elements of finite order for finite monadic Church-Rosser Thue systems
Authors: Otto Friedrich
Published: June 2010

A Thue system

T
T

over

Σ
\Sigma

is said to allow nontrivial elements of finite order, if there exist a word

u
∈

Σ
∗

u \in {\Sigma ^ \ast }

and integers

n
≥
0

n \ge 0

and

k
≥
1

k \ge 1

such that

u
↮

T
∗

λ

u \nleftrightarrow \,_T^ \ast \lambda

and

u

n
+
k

↔

T
∗

u
n

{u^{n + k}} \leftrightarrow \,_T^ \ast {u^n}

. Here the following decision problem is shown to be decidable: Instance. A finite, monadic, Church-Rosser Thue system

T
T

over

Σ
\Sigma

. Question. Does

T
T

allow nontrivial elements of finite order? By a result of Muller and Schupp this implies in particular that given a finite monadic Church-Rosser Thue system

T
T

it is decidable whether the monoid presented by

T
T

is a free group or not.
Projective modules in the category 𝒪_{𝒮}: self-duality
Authors: Irving Ronald S.
Published: June 2010

Given a parabolic subalgebra

p

S

{\mathfrak {p}_S}

of a complex, semisimple Lie algebra

g

\mathfrak {g}

, there is a naturally defined category

O

S

{\mathcal {O}_S}

of

g

\mathfrak {g}

-modules which includes all the

g

\mathfrak {g}

-modules induced from finite-dimensional

p

S

{\mathfrak {p}_S}

-modules. This paper treats the question of whether certain projective modules in

O

S

{\mathcal {O}_S}

are isomorphic to their dual modules. The projectives in question are the projective covers of those simple modules occurring in the socles of generalized Verma modules. Their self-duality is proved in a number of cases, and additional information is obtained on the generalized Verma modules.
The Schur multiplier of a nilpotent group
Authors: Webb Ursula Martin
Published: June 2010

In this paper we obtain upper and lower bounds for the rank of the Schur multiplier of a nilpotent group in terms of the nilpotency class and the number of generators and rank of the derived quotient.
Factorization of diagonally dominant operators on 𝐿₁([0,1],𝑋)
Authors: Andrews Kevin T. , Ward Joseph D.
Published: June 2010

Let

X
X

be a separable Banach space. It is shown that every diagonally dominant invertible operator on

L
1

(
[
0
,

1
]
,

X
)

{L_1}([0,\,1],\,X)

can be factored uniquely as a product of an invertible upper triangular operator and an invertible unit lower triangular operator.
Multipliers on the space of semiperiodic sequences
Authors: Jiménez Manuel Núñez
Published: June 2010

Semiperiodic sequences are defined to be the uniform limit of periodic sequences. They form a space of continuous functions on a compact group

Δ
\Delta

. We study the properties of the Radon measures on

Δ
\Delta

in order to classify the multipliers for the space of semiperiodic sequences, paying special attention to those which can be realized as transference functions of physically constructible filters.
𝐴𝑢𝑡(𝐹)→𝐴𝑢𝑡(𝐹/𝐹”) is surjective for free group 𝐹 of rank ≥4
Authors: Bachmuth Seymour , Mochizuki Horace Y.
Published: June 2010

In this article, it is shown that the group of automorphisms of the free metabelian group

Φ
(
n
)

\Phi (n)

of rank

n
⩾
4

n \geqslant 4

is not only finitely generated but in fact every automorphism of

Φ
(
n
)

\Phi (n)

is induced by an automorphism of the free group of the same rank

n
n

. This contrasts sharply with the authors’ earlier result [4] that any set of generators of the group of automorphisms of the free metabelian group

Φ
(
3
)

\Phi (3)

of rank

3
3

contains infinitely many automorphisms which are not induced by an automorphism of the free group of rank

3
3

.
Brownian motion with polar drift
Authors: Williams R. J.
Published: June 2010

Consider a strong Markov process

X
0

{X^0}

that has continuous sample paths in

R
d

(
d
⩾
2
)

{R^d}(d \geqslant 2)

and the following two properties. (1) Away from the origin

X
0

{X^0}

behaves like Brownian motion with a polar drift given in spherical polar coordinates by

μ
(
θ
)

/

2
r

\mu (\theta )/2r

. Here

μ
\mu

is a bounded Borel measurable function on the unit sphere in

R
d

{R^d}

, with average value

μ
¯

\overline \mu

. (2)

X
0

{X^0}

is absorbed at the origin. It is shown that

X
0

{X^0}

reaches the origin with probability zero or one as

μ
¯

⩾
2
−
d

\overline \mu \geqslant 2 - d

or

>
2
−
d

> 2 - d

. Indeed,

X
0

{X^0}

is transient to

+
∞

+ \infty

if

μ
¯

>
2
−
d

\overline \mu > 2 - d

and null recurrent if

μ
¯

=
2
−
d

\bar \mu = 2 - d

. Furthermore, if

μ
¯

>
2
−
d

\bar \mu > 2 - d

(i.e.,

X
0

{X^0}

reaches the origin), then

X
0

{X^0}

does not approach the origin in any particular direction. Indeed, there is a single Martin boundary point for

X
0

{X^0}

at the origin. The question of the existence and uniqueness of a strong Markov process with continuous sample paths in

R
d

{R^d}

that behaves like

X
0

{X^0}

away from the origin, but spends zero time there (in the sense of Lebesgue measure), is also resolved here.
Algorithmic solution of extremal digraph problems
Authors: Brown W. G. , Erdős P. , Simonovits M.
Published: June 2010

For a given family

L

\mathcal {L}

of digraphs, we study the "extremal" digraphs on

n
n

vertices containing no member of

L

\mathcal {L}

, and having the maximum number of arcs,

ex
⁡
(
n
,

L

)

\operatorname {ex} (n,\mathcal {L})

. We resolve conjectures concerning the set

{

lim

n
→
∞

(
ex
⁡
(
n
,

L

)

/

n
2

)
}

\{ {\lim _{n \to \infty }}(\operatorname {ex} (n,\mathcal {L})/{n^2})\}

as

L

\mathcal {L}

ranges over all possible families, and describe a "finite" algorithm that can determine, for any

L

\mathcal {L}

, all matrices

A
A

for which a sequence

{
A
(
n
)
}

\{ A(n)\}

of "matrix digraphs" is asymptotically extremal (

A
(
n
)

A(n)

contains no member of

L

\mathcal {L}

and has

ex
⁡
(
n
,

L

)
+
o
(

n
2

)

\operatorname {ex} (n,\mathcal {L}) + o({n^2})

arcs as

n
→
∞

n \to \infty

.) Résumé. Pour une famille donnée,

L

\mathcal {L}

, de digraphes, on étudie les digraphes "extrémaux" à

n
n

sommets qui ne contiennent aucun membre de

L

\mathcal {L}

, et qui possèdent le nombre maximal d’arêtes,

ex
⁡
(
n
,

L

)

\operatorname {ex} (n,\mathcal {L})

. On résolue des conjectures qui concernent l’ensemble

{

lim

n
→
∞

(
ex
⁡
(
n
,

L

)

/

n
2

)
}

\{ {\lim _{n \to \infty }}(\operatorname {ex} (n,\mathcal {L})/{n^2})\}

où

L

\mathcal {L}

soit une famille quelconque, et on présente un algorithme "fini" qui peut déterminer, pour chaque

L

\mathcal {L}

, toute matrice

A
A

pour laquelle une suite

{
A
(
n
)
}

\{ A(n)\}

de "digraphes matriciels" est extrémale asymptotiquement (

A
(
n
)

A(n)

ne contient aucun membre de

L

\mathcal {L}

et possède

ex
⁡
(
n
,

L

)
+
o
(

n
2

)

\operatorname {ex} (n,\mathcal {L}) + o({n^2})

arêtes lorsque

n
→
∞

n \to \infty

.)
Joint continuity of division of smooth functions. I. Uniform Lojasiewicz estimates
Authors: Mostow Mark Alan , Shnider Steven
Published: June 2010

In this paper we study the question of the existence of a continuous inverse to the multiplication mapping

(
f
,
g
)
→
(
f
g
,
g
)

(f,g) \to (fg,g)

defined on pairs of

C
∞

{C^\infty }

functions on a manifold

M
M

. Obviously, restrictions must be imposed on the domain of such an inverse. This leads us to the study of a modified problem: Find an appropriate domain for the inverse of

(
f
,
G
)
→
(
f
(
p
∘
G
)
,
G
)

(f,G) \to (f(p \circ G),G)

, where

G
G

is a

C
∞

{C^\infty }

mapping of the manifold

M
M

into an analytic manifold

N
N

and

p
p

is a fixed analytic function on

N
N

. We prove a theorem adequate for application to the study of inverting the mapping

(
A
,
X
)
→
(
A
,
A
X
)

(A,X) \to (A,AX)

, where

X
X

is a vector valued

C
∞

{C^\infty }

function and

A
A

is a square matrix valued

C
∞

{C^\infty }

function on

M
M

whose determinant may vanish on a nowhere dense set.
A dynamical meaning of fractal dimension
Authors: Pelikan Steve
Published: June 2010

When two attractors of a dynamical system have a common basin boundary

B
B

, small changes in initial conditions which lie near

B
B

can result in radically different long-term behavior of the trajectory. A quantitative description of this phenomenon is obtained in terms of the fractal dimension of the basin boundary

B
B

.
Self-maps of loop spaces. I
Authors: Campbell H. E. A. , Peterson F. P. , Selick P. S.
Published: June 2010

In this paper we study self-maps of

Ω
k

S

m
+
1

{\Omega ^k}{S^{m + 1}}

and show that, except for the cases

m
=
1
,

3
,

7

m = 1,\,3,\,7

, or

p
p

and

m
m

, if

f
f

induces an isomorphism on

H

m
+
1
−
k

(

Ω
k

S

m
+
1

;

Z

/

p
Z
)

{H_{m + 1 - k}}({\Omega ^k}{S^{m + 1}};\,Z/pZ)

with

k
>
m

k > m

, then

f

(
p
)

{f_{(p)}}

is a homotopy equivalence.
Quasilinear evolution equations and parabolic systems
Authors: Amann Herbert
Published: June 2010

It is shown that general quasilinear parabolic systems possess unique maximal classical solutions for sufficiently smooth initial values, provided the boundary conditions are “time-independent”. Moreover it is shown that, in the autonomous case, these equations generate local semiflows on appropriate Sobolev spaces. Our results apply, in particular, to the case of prescribed boundary values (Dirichlet boundary conditions).
The trace of an action and the degree of a map
Authors: Gottlieb Daniel Henry
Published: June 2010

Two integer invariants of a fibration are defined: the degree, which generalizes the usual notion, and the trace. These numbers represent the smallest transfers for integral homology which can be constructed for the fibrations. Since every action gives rise to a fibration, we have the trace of an action. A list of properties of this trace is developed. This list immediately gives, in a mechanical way, new proofs and generalizations of theorems of Borsuk-Ulam, P. A. Smith, Conner and Floyd, Bredon, W. Browder, and G. Carlsson.
Rational singularities and almost split sequences
Authors: Auslander Maurice
Published: June 2010

The main aim of this paper is to relate almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup

G
G

of

GL
⁡
(
2
,

C

)

\operatorname {GL} (2,{\mathbf {C}})

, where

C

{\mathbf {C}}

is the complex numbers, is isomorphic to the

A
R

AR

quiver of the reflexive modules of the quotient singularity associated with

G
G

.
Martingale transforms and related singular integrals
Authors: Bañuelos Rodrigo
Published: June 2010

The operators obtained by taking conditional expectation of continuous time martingale transforms are studied, both on the circle

T
T

and on

R

n

{{\mathbf {R}}^n}

. Using a Burkholder-Gundy inequality for vector-valued martingales, it is shown that the vector formed by any number of these operators is bounded on

L
p

(

R

n

)
,

1
>
p
>
∞

{L^p}({{\mathbf {R}}^n}),\,1 > p > \infty

, with constants that depend only on

p
p

and the norms of the matrices involved. As a corollary we obtain a recent result of Stein on the boundedness of the Riesz transforms on

L
p

(

R

n

)
,

1
>
p
>
∞

{L^p}({{\mathbf {R}}^n}),\,1 > p > \infty

, with constants independent of

n
n

.
Simple homotopy type of finite 2-complexes with finite abelian fundamental group
Authors: Latiolais M. Paul
Published: June 2010

Theorem 1. Let

K
K

be a finite

2
2

-dimensional

C
W

CW

-complex with

π
1

(
K
)

{\pi _1}(K)

finite and abelian. Then every element of the Whitehead group of

K
K

is realizable as the torsion of a self-homotopy equivalence on

K
K

. Theorem 2. Homotopy equivalence and simple homotopy equivalence are the same for finite

2
2

-dimensional

C
W

CW

-complexes with finite abelian fundamental groups.
Almost Mizohata operators
Authors: Hanges Nicholas
Published: June 2010

The range of every almost Mizohata operator is characterized via an integral formula.
The Stefan problem with heating: appearance and disappearance of a mushy region
Authors: Bertsch M. , de Mottoni P. , Peletier L. A.
Published: June 2010

We consider a material which is initially in the solid state and then, due to heating, starts to melt. We describe the appearance of a so-called mushy region, i.e., a region in which the material is in neither a solid nor a liquid state. The main result is that after a finite time the mushy region has disappeared and only the solid and the liquid phases have remained.
Potentials producing maximally sharp resonances
Authors: Harrell Evans M. , Svirsky Roman
Published: June 2010

We consider quantum-mechanical potentials consisting of a fixed background plus an additional piece constrained only by having finite height and being supported in a given finite region in dimension

d
⩽
3

d \leqslant 3

. We characterize the potentials in this class that produce the sharpest resonances. In the one-dimensional or spherically symmetric specialization, a quite detailed description is possible. The maximally sharp resonances that we find are, roughly speaking, caused by barrier confinement of a metastable state, although in some situations they call for interactions in the interior of the confining barrier as well.
Spectral theory of the linearized Vlasov-Poisson equation
Authors: Degond Pierre
Published: June 2010

We study the spectral theory of the linearized Vlasov-Poisson equation, in order to prove that its solution behaves, for large times, like a sum of plane waves. To obtain such an expansion involving damped waves, we must find an analytical extension of the resolvent of the equation. Then, the poles of this extension are no longer eigenvalues and must be interpreted as eigenmodes, associated to “generalized eigenfunctions” which actually are linear functionals on a Banach space of analytic functions.
The Auslander-Reiten quiver of a simple curve singularity
Authors: Dieterich Ernst , Wiedemann Alfred
Published: June 2010

With any simple curve singularity (plane, complex, affine-algebraic) of Dynkin-type

Δ
\Delta

we associate the category of all finitely generated torsionfree modules over its complete local ring. For each of these module categories we calculate the Auslander-Reiten quiver. We suggest the construction of the “twisted quiver” of a quiver with involution and valuation of arrows which gives rise to a (purely combinatorial) one-to-one correspondence between the Auslander-Reiten quiver and the Dynkin diagram

Δ
\Delta

.
Unitary quasilifting: applications
Authors: Flicker Yuval Z.
Published: June 2010

Let

U
(
3
)

U(3)

be the quasi-split unitary group in three variables defined using a quadratic extension

E

/

F

E/F

of number fields. Complete local and global results are obtained for the

σ
\sigma

-endo-(unstable) lifting from

U
(
2
)

U(2)

to

GL

(
3
,

E
)

{\text {GL}}(3,\,E)

. This is used to establish quasi-(endo-)lifting for automorphic forms from

U
(
2
)

U(2)

to

U
(
3
)

U(3)

by means of base change from

U
(
3
)

U(3)

to

GL

(
3
,

E
)

{\text {GL}}(3,\,E)

. Base change quasi-lifting is also proven. Continuing the work of

[

I

]

\left [ {\mathbf {I}} \right ]

, the exposition is elementary, and uses only a simple form of an identity of trace formulas, and base change transfer of orbital integrals of spherical functions.
Resonance and quasilinear ellipticity
Authors: Shapiro Victor L.
Published: June 2010

Two resonance-type existence theorems for periodic solutions of second order quasilinear elliptic partial differential equations are established. The first theorem is a best possible result, and the second theorem presents conditions which are both necessary and sufficient.
Functional equations for character series associated with 𝑛×𝑛 matrices
Authors: Formanek Edward
Published: June 2010

Let

A
A

be either the ring of invariants or the trace ring of

r
r

generic

n
×
n

n \times n

matrices. Then

A
A

has a character series

χ
(
A
)

\chi (A)

which is a symmetric rational function of commuting variables

x
1

,
…
,

x
r

{x_1}, \ldots ,{x_r}

. The main result is that if

r
≥

n
2

r \geq {n^2}

, then

χ
(
A
)

\chi (A)

satisfies the functional equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi left-parenthesis upper A right-parenthesis left-parenthesis x 1 Superscript negative 1 Baseline comma ellipsis comma x Subscript r Superscript negative 1 Baseline right-parenthesis equals left-parenthesis negative 1 right-parenthesis Superscript d Baseline left-parenthesis x 1 midline-horizontal-ellipsis x Subscript r Baseline right-parenthesis Superscript n squared Baseline chi left-parenthesis upper A right-parenthesis left-parenthesis x 1 comma ellipsis comma x Subscript r Baseline right-parenthesis">

χ
(
A
)
(

x
1

−
1

,
…
,

x
r

−
1

)
=

(
−
1

)
d

(

x
1

⋯

x
r

)

n
2

χ
(
A
)
(

x
1

,
…
,

x
r

)

\chi (A)(x_1^{ - 1}, \ldots ,x_r^{ - 1}) = {( - 1)^d}{({x_1} \cdots {x_r})^{{n^2}}}\chi (A)({x_1}, \ldots ,{x_r})

\]
, where

d
d

is the Krull dimension of

A
A

.
Weak type estimates for Bochner-Riesz spherical summation multipliers
Authors: Chanillo Sagun , Muckenhoupt Benjamin
Published: June 2010

We consider the Bochner-Riesz multiplier
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper T Subscript delta Baseline f With caret left-parenthesis xi right-parenthesis equals left-parenthesis 1 minus StartAbsoluteValue xi EndAbsoluteValue squared right-parenthesis Superscript delta Baseline plus ModifyingAbove f With caret left-parenthesis xi right-parenthesis comma delta greater-than 0 comma">

T
δ

f

^

(
ξ
)
=

(
1
−

|
ξ
|

2

)
δ

+

f
^

(
ξ
)
,

δ
>
0
,

\widehat {{T_\delta }f}(\xi ) = {(1 - {\left | \xi \right |^2})^\delta } + \hat f(\xi ),\qquad \delta > 0,

\]
where

^

\widehat {}

denotes the Fourier transform. It is shown that the multiplier operator

T
δ

{T_\delta }

is weak type

(

p
0

,

p
0

)

({p_0},\,{p_0})

acting on

L

p
0

(

R

n

)

{L^{p0}}({{\mathbf {R}}^n})

radial functions, where

p
0

{p_0}

is the critical value

2
n

/

(
n
+
1
+
2
δ
)

2n/(n + 1 + 2\delta )

.
Trellises formed by stable and unstable manifolds in the plane
Authors: Easton Robert W.
Published: June 2010

A trellis is the figure formed by the stable and unstable manifolds of a hyperbolic periodic point of a diffeomorphism of a

2
2

-manifold. This paper describes and classifies some trellises. The set of homoclinic points is linearly ordered as a subset of the stable manifold and again as a subset of the unstable manifold. Each homoclinic point is assigned a type number which is constant on its orbit. Combinatorial properties of trellises are studied using type numbers and the pair of linear orderings. Trellises are important because their closures in some cases are strange attractors and in other cases are ergodic zones.
Congruences on regular semigroups
Authors: Pastijn Francis , Petrich Mario
Published: June 2010

Let

S
S

be a regular semigroup and let

ρ
\rho

be a congruence relation on

S
S

. The kernel of

ρ
\rho

, in notation

ker
⁡
ρ

\ker \rho

, is the union of the idempotent

ρ
\rho

-classes. The trace of

ρ
\rho

, in notation

tr

ρ

\operatorname {tr}\,\rho

, is the restriction of

ρ
\rho

to the set of idempotents of

S
S

. The pair

(
ker
⁡
ρ
,
tr

ρ
)

(\ker \rho ,\operatorname {tr}\,\rho )

is said to be the congruence pair associated with

ρ
\rho

. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple

(
(
ρ
∨

L

)

/

L

,
ker
⁡
ρ
,
(
ρ
∨

R

)

/

R

)

((\rho \vee \mathcal {L})/\mathcal {L},\ker \rho ,(\rho \vee \mathcal {R})/\mathcal {R})

is said to be the congruence triple associated with

ρ
\rho

. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of

S
S

. For congruence relations

ρ
\rho

and

θ
\theta

, put

ρ

T
l

θ

[
ρ

T
r

θ
,
ρ
T
θ
]

\rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]

if and only if

ρ
∨

L

=
θ
∨

L

[
ρ
∨

R

=
θ
∨

R

,
tr
⁡
ρ
=
tr
⁡
θ
]

\rho \vee \mathcal {L} = \theta \vee \mathcal {L}\;[\rho \vee \mathcal {R} = \theta \vee \mathcal {R},\operatorname {tr}\rho = \operatorname {tr}\theta ]

. Then

T
l

,

T
r

{T_l},{T_r}

and

T
T

are complete congruences on the congruence lattice of

S
S

and

T
=

T
l

∩

T
r

T = {T_l} \cap {T_r}

.
Quantization and Hamiltonian 𝐺-foliations
Authors: Pukanszky L.
Published: June 2010

As it was recognized twenty five years ago by A. A. Kirillov, in the unitary representation theory of nilpotent Lie groups a crucial role is played by orbits of the coadjoint representation. B. Kostant noted that, for any connected Lie group, these orbits admit a symplectic structure and lend themselves to an intrinsic characterization. The present author later observed, that already for the purposes of unitary representation theory of solvable Lie groups, this concept has to be enlarged and replaced by that of a generalized orbit. One objective of this paper is their intrinsic characterization. Other results prepare the way for the geometric construction of the corresponding unitary representations, to be developed later.
Isometries for the Legendre-Fenchel transform
Authors: Attouch Hédy , Wets Roger J.-B.
Published: June 2010

It is shown that on the space of lower semicontinuous convex functions defined on

R
n

{R^n}

, the conjugation map—the Legendre-Fenchel transform—is an isometry with respect to some metrics consistent with the epi-topology. We also obtain isometries for the infinite dimensional case (Hilbert space and reflexive Banach space), but this time they correspond to topologies finer than the Moscoepi-topology.
Paths and cycles in tournaments
Authors: Thomason Andrew
Published: June 2010

Sufficient conditions are given for the existence of an oriented path with given end vertices in a tournament. As a consequence a conjecture of Rosenfeld is established. This states that if

n
n

is large enough, then every non-strongly oriented cycle of order

n
n

is contained in every tournament of order

n
n

.
“Almost” implies “near”
Authors: Anderson Robert M.
Published: June 2010

We formulate a formal language in which it is meaningful to say that an object almost satisfies a property. We then show that any object which almost satisfies a property is near an object which exactly satisfies the property. We show how this principle can be used to prove existence theorems. We give an example showing how one may strengthen the statement to give information about the relationship between the amount by which the object fails to satisfy the property and the distance to the nearest object which satisfies the property. Examples are given concerning commuting matrices, additive sequences, Brouwer fixed points, competitive equilibria, and differential equations.
On excursions of reflecting Brownian motion
Authors: Hsu Pei
Published: June 2010

We discuss the properties of excursions of reflecting Brownian motion on a bounded smooth domain in

R
d

{R^d}

and give a procedure for constructing the process from the excursions and the boundary process. Our method is computational and can be applied to general diffusion processes with reflecting type boundary conditions on compact manifolds.
Hypothesis testing in integral geometry
Authors: Waksman Peter
Published: June 2010

Probability distributions are defined relative to a fixed plane domain and are calculated explicitly when the domain is a union of coordinate rectangles. The theory of approximating step functions by the resulting special functions gives an interpretation of the problem of guessing a domain given a random sample of observations.
On the homology of associative algebras
Authors: Anick David J.
Published: June 2010

We present a new free resolution for

k
k

as an

G
G

-module, where

G
G

is an associative augmented algebra over a field

k
k

. The resolution reflects the combinatorial properties of

G
G

.
Weighted inequalities for the one-sided Hardy-Littlewood maximal functions
Authors: Sawyer E.
Published: June 2010

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Superscript plus Baseline f left-parenthesis x right-parenthesis equals sup Underscript h greater-than 0 Endscripts left-parenthesis 1 slash h right-parenthesis integral Subscript x Superscript x plus h Baseline StartAbsoluteValue f left-parenthesis t right-parenthesis EndAbsoluteValue d t">

M
+

f
(
x
)
=

sup

h
>
0

(
1

/

h
)

∫
x

x
+
h

|

f
(
t
)

|

d
t

{M^ + }f(x) = {\sup _{h > 0}}(1/h)\int _x^{x + h} {|f(t)|\,dt}

denote the one-sided maximal function of Hardy and Littlewood. For

w
(
x
)
⩾
0

w(x) \geqslant 0

on

R
R

and

1
>
p
>
∞

1 > p > \infty

, we show that

M
+

{M^ + }

is bounded on

L
p

(
w
)

{L^p}(w)

if and only if

w
w

satisfies the one-sided

A
p

{A_p}

condition:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A Subscript p Superscript plus Baseline right-parenthesis left-bracket StartFraction 1 Over h EndFraction integral Subscript a minus h Superscript a Baseline w left-parenthesis x right-parenthesis d x right-bracket left-bracket StartFraction 1 Over h EndFraction integral Subscript a Superscript a plus h Baseline w left-parenthesis x right-parenthesis Superscript negative 1 slash left-parenthesis p minus 1 right-parenthesis Baseline d x right-bracket Superscript p minus 1 Baseline less-than-or-slanted-equals upper C">

(

A
p
+

)

[

1
h

∫

a
−
h

a

w
(
x
)
d
x

]

[

1
h

∫
a

a
+
h

w

(
x
)

−
1

/

(
p
−
1
)

d
x

]

p
−
1

⩽
C

\left ( {A_p^ + } \right )\qquad \left [ {\frac {1} {h}\int _{a - h}^a {w(x)dx} } \right ]{\left [ {\frac {1} {h}\int _a^{a + h} {w{{(x)}^{ - 1/(p - 1)}}dx} } \right ]^{p - 1}} \leqslant C

\]
for all real

a
a

and positive

h
h

. If in addition

v
(
x
)
⩾
0

v(x) \geqslant 0

and

σ
=

v

−
1

/

(
p
−
1
)

\sigma = {v^{ - 1/(p - 1)}}

,then

M
+

{M^ + }

is bounded from

L
p

(
v
)

{L^p}(v)

to

L
p

(
w
)

{L^p}(w)

if and only if
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="integral Underscript upper I Endscripts left-bracket upper M Superscript plus Baseline left-parenthesis chi Subscript upper I Baseline sigma right-parenthesis right-bracket Superscript p Baseline w less-than-or-slanted-equals upper C integral Underscript upper I Endscripts sigma greater-than normal infinity">

∫
I

[

M
+

(

χ
I

σ
)
]

p

w
⩽
C

∫
I

σ
>
∞

\int _I {{{[{M^ + }({\chi _I}\sigma )]}^p}w \leqslant C\int _I {\sigma > \infty } }

\]
for all intervals

I
=
(
a
,
b
)

I = (a,b)

such that

∫

−
∞

a

w
>
0

\int _{ - \infty }^a {w > 0}

. The corresponding weak type inequality is also characterized. Further properties of

A
p
+

A_p^ +

weights, such as

A
p
+

⇒

A

p
−
ε

+

A_p^ + \Rightarrow A_{p - \varepsilon }^ +

and

A
p
+

=
(

A
1
+

)

(

A
1
−

)

1
−
p

A_p^ + = (A_1^ + ){(A_1^ - )^{1 - p}}

, are established.
À propos de “wedges” et d’“edges”, et de prolongements holomorphes
Authors: Rosay Jean-Pierre
Published: June 2010

Holomorphic extensions in wedges of continuous functions defined on edges, which are extensions in the distributional sense, are shown to be genuine continuous extensions, and a

C

1

{\mathcal {C}^1}

version of the edge of the wedge theorem is proved.
Equivariant minimal immersions of 𝑆² into 𝑆^{2𝑚}(1)
Authors: Ejiri Norio
Published: June 2010

We classify the directrix curves associated with equivariant minimal immersions of

S
2

{S^2}

into

S

2
m

(
1
)

{S^{2m}}(1)

and obtain some applications.
The blow-up boundary for nonlinear wave equations
Authors: Caffarelli Luis A. , Friedman Avner
Published: June 2010

Consider the Cauchy problem for a nonlinear wave equation

◻

u
=
F
(
u
)

\square u = F(u)

in

N
N

space dimensions,

N
⩽
3

N \leqslant 3

, with

F
F

superlinear and nonnegative. It is well known that, in general, the solution blows up in finite time. In this paper it is shown, under some assumptions on the Cauchy data, that the blow-up set is a space-like surface

t
=
ϕ
(
x
)

t = \phi (x)

with

ϕ
(
x
)

\phi (x)

continuously differentiable.
Countable-dimensional universal sets
Authors: Pol Roman
Published: June 2010

The main results of this paper are a construction of a countable union of zero dimensional sets in the Hilbert cube whose complement does not contain any subset of finite dimension

n
⩾
1

n \geqslant 1

(Theorem 2.1, Corollary 2.3) and a construction of universal sets for the transfinite extension of the Menger-Urysohn inductive dimension (Theorem 2.2, Corollary 2.4).
On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type
Authors: Ni Wei-Ming , Takagi Izumi
Published: June 2010

We derive a priori estimates for positive solutions of the Neumann problem for some semilinear elliptic systems (i.e., activator-inhibitor systems in biological pattern formation theory), as well as for semilinear single equations related to such systems. By making use of these a priori estimates, we show that under certain assumptions, there is no positive nonconstant solutions for single equations or for activator-inhibitor systems when the diffusion coefficient (of the activator, in the case of systems) is sufficiently large; we also study the existence of nonconstant solutions for specific domains.
A characterization and another construction of Janko’s group 𝐽₃
Authors: Weiss Richard
Published: June 2010

Graphs

Γ
\Gamma

with the following properties are classified: (i)

Γ
\Gamma

is

(
G
,
s
)

(G,s)

-transitive for some

s
⩾
4

s \geqslant 4

and some group

G
⩽
aut
⁡
(
Γ
)

G \leqslant \operatorname {aut} (\Gamma )

such that each vertex stabilizer in

G
G

is finite, (ii)

s
⩾
(
g
−
1
)

/

2

s \geqslant (g - 1)/2

, where

g
g

is the girth of

Γ
\Gamma

, and (ii)

Γ
\Gamma

is connected. A new construction of

J
3

{J_3}

is given.