bluuebunny/crossref_metadata_2025 huggingface dataset:
A general approach to the optimality of minimum distance estimators
Authors: Millar P. W.
Published: June 2010

Let

Θ
\Theta

be an open subset of a separable Hilbert space, and

ξ
n

(
θ
)

{\xi _n}(\theta )

,

θ
∈
Θ

\theta \in \Theta

, a sequence of stochastic processes with values in a (different) Hilbert space

B
B

. This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates"

θ
^

n

{\hat \theta _n}

defined by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf Underscript theta Endscripts StartAbsoluteValue xi Subscript n Baseline left-parenthesis theta right-parenthesis EndAbsoluteValue equals StartAbsoluteValue xi Subscript n Baseline left-parenthesis ModifyingAbove theta With caret Subscript n Baseline right-parenthesis EndAbsoluteValue">

inf
θ

|

ξ
n

(
θ
)

|

=

|

ξ
n

(

θ
^

n

)

|

{\inf _\theta }|{\xi _n}(\theta )| = |{\xi _n}({\hat \theta _n})|

, where

|

⋅

|

| \cdot |

is the norm of

B
B

. The abstract results are applied to study optimality and asymptotic normality of procedures in a number of important practical problems, including simple regression, spectral function estimation, quantile function methods, min-chi-square methods, min-Hellinger methods, minimum distance methods based on

M
M

-functionals, and so forth. The results unify several studies in the literature, but most of the

LAM

{\text {LAM}}

results are new. From the point of view of applications, the entire paper is a sustained essay concerning the problem of fitting data with a reasonable, but relatively simple, model that everyone knows cannot be exact.
Inverse producing extension of a Banach algebra which eliminates the residual spectrum of one element
Authors: Read C. J.
Published: June 2010

If

A
A

is a commutative unital Banach algebra and

G
⊂
A

G \subset A

is a collection of nontopological zero divisors, the question arises whether we can find an extension

A
′

A\prime

of

A
A

in which every element of

G
G

has an inverse. Shilov [1] proved that this was the case if

G
G

consisted of a single element, and Arens [2] conjectures that it might be true for any set

G
G

. In [3], Bollobás proved that this is not the case, and gave an example of an uncountable set

G
G

for which no extension

A
′

A\prime

can contain inverses for more than countably many elements of

G
G

. Bollobás proved that it was possible to find inverses for any countable

G
G

, and gave best possible bounds for the norms of the inverses in [4]. In this paper, it is proved that inverses can always be found if the elements of

G
G

differ only by multiples of the unit; that is, we can eliminate the residual spectrum of one element of

A
A

. This answers the question posed by Bollobás in [5].
Bowen-Ruelle measures for certain piecewise hyperbolic maps
Authors: Young Lai-Sang
Published: June 2010

We consider a class of piecewise

C
2

{C^2}

Lozi-like maps and prove the existence of invariant measures with absolutely continuous conditional measures on unstable manifolds
Mixed projection inequalities
Authors: Lutwak Erwin
Published: June 2010

A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its

i
i

th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.
Fractional integrals on weighted 𝐻^{𝑝} and 𝐿^{𝑝} spaces
Authors: Strömberg Jan-Olov , Wheeden Richard L.
Published: June 2010

We study the two weight function problem

∥

I
α

f

∥

H
u
q

⩽
c
∥
f

∥

H
v
p

,
0
>
p
⩽
q
>
∞

\parallel {I_\alpha }f{\parallel _{H_u^q}} \leqslant c\parallel f{\parallel _{H_v^p}},0 > p \leqslant q > \infty

, for fractional integrals on Hardy spaces. If

u
u

and

v
v

satisfy the doubling condition and

0
>
p
⩽
1

0 > p \leqslant 1

, we obtain a necessary and sufficient condition for the norm inequality to hold. If

1
>
p
>
∞

1 > p > \infty

we obtain a necessary condition and a sufficient condition, and show these are the same under various additional conditions on

u
u

and

v
v

. We also consider the corresponding problem for

L
u
q

L_u^q

and

L
v
p

L_v^p

, and obtain a necessary and sufficient condition in some cases.
Properties of center manifolds
Authors: Sijbrand Jan
Published: June 2010

The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties. Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than

c
1

exp
⁡
(

c
2

x

−
1

)

{c_1}\exp ({c_2}{x^{ - 1}})

with

c
1

{c_1}

and

c
2

{c_2}

constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge. The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost. Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.
Translation representations for automorphic solutions of the wave equation in non-Euclidean spaces; the case of finite volume
Authors: Lax Peter D. , Phillips Ralph S.
Published: June 2010

Let

Γ
\Gamma

be a discrete subgroup of automorphisms of

H

n

{{\mathbf {H}}^n}

, with fundamental polyhedron of finite volume, finite number of sides, and

N
N

cusps. Denote by

Δ
Γ

{\Delta _\Gamma }

the Laplace-Beltrami operator acting on functions automorphic with respect to

Γ
\Gamma

. We give a new short proof of the fact that

Δ
Γ

{\Delta _\Gamma }

has absolutely continuous spectrum of uniform multiplicity

N
N

on

(
−
∞
,

(
(
n
−
1
)

/

2

)
2

)

( - \infty ,{((n - 1)/2)^2})

, plus a standard discrete spectrum. We show that this property of the spectrum is unchanged under arbitrary perturbation of the metric on a compact set. Our method avoids Eisenstein series entirely and proceeds instead by constructing explicitly a translation representation for the associated wave equation.
Regular linear algebraic monoids
Authors: Putcha Mohan S.
Published: June 2010

In this paper we study connected regular linear algebraic monoids. If

ϕ
:

G
0

→

GL

(
n
,
K
)

\phi :{G_0} \to {\text {GL}}(n,K)

is a representation of a reductive group

G
0

{G_0}

, then the Zariski closure of

K
ϕ
(

G
0

)

K\phi ({G_0})

in

M

n

(
K
)

{\mathcal {M}_n}(K)

is a connected regular linear algebraic monoid with zero. In

§
2

\S 2

we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are

V
V

-regular. In

§
3

\S 3

we show that when the ideals are linearly ordered, the group of units is nearly simple of type

A
l

,

B
l

,

C
l

,

F
4

or

G
2

{A_l},{B_l},{C_l},{F_4}\;{\text {or}}\;{G_2}

. In

§
4

\S 4

, conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.
Normal subgroups of doubly transitive automorphism groups of chains
Authors: Ball Richard N. , Droste Manfred
Published: June 2010

We characterize the structure of the normal subgroup lattice of22-transitive automorphism groupsA(Ω)A(\Omega )of infinite chains(Ω,⩽)(\Omega , \leqslant )by the structure of the Dedekind completion(Ω¯,⩽)(\bar \Omega , \leqslant )of the chain(Ω,⩽)(\Omega , \leqslant ). As a consequence we obtain various group-theoretical results on the normal subgroups ofA(Ω)A(\Omega ), including that any proper subnormal subgroup ofA(Ω)A(\Omega )is indeed normal and contained in a maximal proper normal subgroup ofA(Ω)A(\Omega ), and thatA(Ω)A(\Omega )has precisely55normal subgroups if and only if the coterminality of the chain(Ω,⩽)(\Omega , \leqslant )is countable.
Changing the depth of an ordered set by decomposition
Authors: Milner E. C. , Prikry K.
Published: June 2010

The depth of a partially ordered set

⟨
P
,
>
⟩

\langle P, > \rangle

is the smallest ordinal

γ
\gamma

such that

⟨
P
,
>
⟩

\langle P, > \rangle

does not embed

γ
∗

{\gamma ^\ast }

. The width of

⟨
P
,
>
⟩

\langle P, > \rangle

is the smallest cardinal number

μ
\mu

such that there is no antichain of size

μ
+
1

\mu + 1

in

P
P

. We show that if

γ
>
ω

\gamma > \omega

and

γ
\gamma

is not an infinite successor cardinal, then any partially ordered set of depth

γ
\gamma

can be decomposed into

cf
⁡
(

|

γ

|

)

\operatorname {cf}(|\gamma |)

parts so that the depth of each part is strictly less than

γ
\gamma

. If

γ
=
ω

\gamma = \omega

or if

γ
\gamma

is an infinite successor cardinal, then for any infinite cardinal

λ
\lambda

there is a linearly ordered set of depth

γ
\gamma

such that for any

λ
\lambda

-decomposition one of the parts has the same depth

γ
\gamma

. These results are used to solve an analogous problem about width. It is well known that, for any cardinal

λ
\lambda

, there is a partial order of width

ω
\omega

which cannot be split into

λ
\lambda

parts of finite width. We prove that, for any cardinal

λ
\lambda

and any infinite cardinal

ν
\nu

, there is a partial order of width

ν
+

{\nu ^ + }

which cannot be split into

λ
\lambda

parts of smaller width.
Local vanishing properties of solutions of elliptic and parabolic quasilinear equations
Authors: Díaz J. Ildefonso , Véron Laurent
Published: June 2010

We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation

−
div

A
(
x
,
u
,
D
u
)
+
B
(
x
,
u
,
D
u
)
=
0

- \operatorname {div}\;A(x,u,Du) + B(x,u,Du) = 0

and of the parabolic equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="partial-differential psi left-parenthesis u right-parenthesis slash partial-differential t minus d i v script upper A left-parenthesis t comma x comma u comma upper D u right-parenthesis plus script upper B left-parenthesis t comma x comma u comma upper D u right-parenthesis equals 0">

∂
ψ
(
u
)

/

∂
t
−
div

A

(
t
,
x
,
u
,
D
u
)
+

B

(
t
,
x
,
u
,
D
u
)
=
0

\partial \psi (u)/\partial t - \operatorname {div}\;\mathcal {A}(t,x,u,Du) + \mathcal {B}(t,x,u,Du) = 0

. The results are obtained without any assumption of monotonicity on

A
A

,

B
B

,

A

\mathcal {A}

and

B

\mathcal {B}

.
A “Tits-alternative” for subgroups of surface mapping class groups
Authors: McCarthy John
Published: June 2010

It has been observed that surface mapping class groups share various properties in common with the class of linear groups (e.g.,

[

B
L
M

,

H

]

[\mathbf {BLM},\,\mathbf {H}]

). In this paper, the known list of such properties is extended to the “Tits-Alternative”, a property of linear groups established by J. Tits

[

T

]

[\mathbf {T}]

. In fact, we establish that every subgroup of a surface mapping class group is either virtually abelian or contains a nonabelian free group. In addition, in order to establish this result, we develop a theory of attractors and repellers for the action of surface mapping classes on Thurston’s projective lamination spaces

[

T
h
1

]

[\mathbf {Th1}]

. This theory generalizes results known for pseudo-Anosov mapping classes

[

F
L
P

]

[\mathbf {FLP}]

.
Saturation of the closed unbounded filter on the set of regular cardinals
Authors: Jech Thomas J. , Woodin W. Hugh
Published: June 2010

For any

α
>

κ
+

\alpha > {\kappa ^ + }

, the following are equiconsistent: (a)

κ
\kappa

is measurable of order

α
\alpha

, (b)

κ
\kappa

is

α
\alpha

-Mahlo and the filter

C

[
Reg
]

{\mathbf {C}}[\operatorname {Reg} ]

is saturated.
Les groupes 𝜔-stables de rang fini
Authors: Lascar Daniel
Published: June 2010

We prove that a group

G
G

which is

ω
\omega

-stable of finite Morley rank is nonmultidimensional. If moreover it is connected and does not have any infinite normal abelian definable subgroup, then it is isomorphic to

Π

H
i

/

K

\Pi {H_i}/K

, where the

H
i

{H_i}

are

ω
1

{\omega _1}

-categorical groups and

K
K

is a finite group.
A Dowker product
Authors: Bešlagić Amer
Published: June 2010

♢
\diamondsuit

implies that there is a (normal) countably paracompact space

X
X

such that

X
2

{X^2}

is normal and not countably paracompact.
An irreducible representation of a symmetric star algebra is bounded
Authors: Bhatt Subhash J.
Published: June 2010

A

∗

{\ast }

-algebra

A
A

is called symmetric if

(
1
+

x

∗

x
)

(1 + {x^{\ast }}x)

is invertible in

A
A

for each

x
x

in

A
A

. An irreducible hermitian representation of a symmetric

∗

{\ast }

-algebra

A
A

maps

A
A

onto an algebra of bounded operators.
Combinatorial lower bound arguments for deterministic and nondeterministic Turing machines
Authors: Maass Wolfgang
Published: June 2010

We introduce new techniques for proving quadratic lower bounds for deterministic and nondeterministic11-tape Turing machines (all considered Turing machines have an additional one-way input tape). In particular, we derive for the simulation of22-tape Turing machines by11-tape Turing machines an optimal quadratic lower bound in the deterministic case and a nearly optimal lower bound in the nondeterministic case. This answers the rather old question whether the computing power of the considered types of Turing machines is significantly increased when more than one tape is used (problem Nos. 1 and 7 in the list of Duris, Galil, Paul, Reischuk [3]). Further, we demonstrate a substantial superiority of nondeterminism over determinism and of co-nondeterminism over nondeterminism for11-tape Turing machines.
A rearranged good 𝜆 inequality
Authors: Bagby Richard J. , Kurtz Douglas S.
Published: June 2010

Let

T
f

Tf

be a maximal Calderón-Zygmund singular integral,

M
f

Mf

the Hardy-Littlewood maximal function, and

w
w

an

A
∞

{A_\infty }

weight. We replace the “good

λ
\lambda

” inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w left-parenthesis StartSet x colon upper T f left-parenthesis x right-parenthesis greater-than 2 lamda and upper M f left-parenthesis x right-parenthesis less-than-or-equal-to epsilon lamda EndSet right-parenthesis less-than-or-equal-to upper C left-parenthesis epsilon right-parenthesis w left-parenthesis StartSet x colon upper T f left-parenthesis x right-parenthesis greater-than lamda EndSet right-parenthesis">

w

(

{
x
:

T
f
(
x
)
>
2
λ

and

M
f
(
x
)
≤
ε
λ
}

)

≤
C
(
ε
)
w

(

{
x
:

T
f
(
x
)
>
λ
}

)

w\left ( {\{ x:\,Tf(x) > 2\lambda \,{\text {and}}\,Mf(x) \leq \varepsilon \lambda \} } \right ) \leq C(\varepsilon )w\left ( {\{ x:\,Tf(x) > \lambda \} } \right )

\]
by the rearrangement inequality
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper T f right-parenthesis Subscript w Superscript asterisk Baseline left-parenthesis t right-parenthesis less-than-or-equal-to upper C left-parenthesis upper M f right-parenthesis Subscript w Superscript asterisk Baseline left-parenthesis t slash 2 right-parenthesis plus left-parenthesis upper T f right-parenthesis Subscript w Superscript asterisk Baseline left-parenthesis 2 t right-parenthesis">

(
T
f

)
w
∗

(
t
)
≤
C
(
M
f

)
w
∗

(
t

/

2
)
+
(
T
f

)
w
∗

(
2
t
)

(Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t)

\]
and show that it gives better estimates for

T
f

Tf

. In particular, we obtain best possible weighted

L
p

{L^p}

bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for

(
T
f

)
∗

{(Tf)^ \ast }

.
The balanced-projective dimension of abelian 𝑝-groups
Authors: Fuchs L. , Hill P.
Published: June 2010

The balanced-projective dimension of every abelian

p
p

-group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for

λ
=

ω
n

\lambda = {\omega _n}

that every

p
λ

{p^\lambda }

-high subgroup of a

p
p

-group

G
G

has balanced-projective dimension exactly

n
n

whenever

G
G

has cardinality

ℵ
n

{\aleph _n}

but

p
λ

G
≠
0

{p^\lambda }G \ne 0

. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished.
McKay quivers and extended Dynkin diagrams
Authors: Auslander Maurice , Reiten Idun
Published: June 2010

Let

k
k

be an algebraically closed field and

G
G

a finite nontrivial group whose order is not divisible by the characteristic of

k
k

. Associated with an

m
m

-dimensional representation of

G
G

is the McKay quiver, whose vertices correspond to the irreducible representations of

G
G

. We show that if

m
=
2

m = 2

, then the underlying graph of the separated McKay quiver is a finite union of extended Dynkin diagrams.
Symmetric semicontinuity implies continuity
Authors: Uher Jaromir
Published: June 2010

The main result of this paper shows that, for any function, symmetric semicontinuity on a measurable set

E
E

implies continuity a.e. in

E
E

and, similarly, that symmetric semicontinuity on a set residual in

R
R

implies continuity on a set residual in

R
R

. These propositions are used to prove more precise versions of the fundamental connections between symmetric and ordinary differentiability.
Principal lines on surfaces immersed with constant mean curvature
Authors: Gutiérrez C. , Sotomayor J.
Published: June 2010

Configurations of lines of principal curvature on constant mean curvature immersed surfaces are studied. Analytical models for these configurations near general isolated umbilical points and particular types of ends are found. From the existence of transversal invariant measures for the foliations by principal lines, established here, follows that the union of recurrent lines of principal curvature is an open set. Examples illustrating all possible cases are given.
Growth rates and critical exponents of classes of binary combinatorial geometries
Authors: Kung Joseph P. S.
Published: June 2010

We prove that a binary geometry of rank

n

(
n
⩾
2
)

n\;(n \geqslant 2)

not containing

M
(

K
5

)

M({K_5})

and

F
7

{F_7}

(respectively,

M
(

K
5

)

M({K_5})

and

C

10

{C_{10}}

) as a minor has at most

3
n
−
3

3n - 3

(respectively,

4
n
−
5

4n - 5

) points. Here,

M
(

K
5

)

M({K_5})

is the cycle geometry of the complete graph on five vertices,

F
7

{F_7}

the Fano plane, and

C

10

{C_{10}}

a certain rank

4
4

ten-point geometry containing the dual Fano plane

F
7

∗

F_7^{\ast }

as a minor. Our technique is elementary and uses the notion of a bond graph. From these results, we deduce upper bounds on the critical exponents of these geometries.
Potent axioms
Authors: Foreman Matthew
Published: June 2010

This paper suggests alternatives to the ordinary large cardinal axioms of set theory. These axioms can be viewed as generalizations of large cardinals and exhibit many of the same phenomena. They are shown to imply the G.C.H., every set of reals in

L
(

R

)

L({\mathbf {R}})

is Lebesgue measurable, and various results in combinatorics, algebra and model theory.
On sieved orthogonal polynomials. III. Orthogonality on several intervals
Authors: Ismail Mourad E. H.
Published: June 2010

We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x colon minus 2 StartRoot c EndRoot slash left-parenthesis 1 plus c right-parenthesis less-than-or-slanted-equals upper T Subscript k Baseline left-parenthesis x right-parenthesis less-than-or-slanted-equals 2 StartRoot c EndRoot slash left-parenthesis 1 plus c right-parenthesis EndSet">

{
x
:
−
2

c

/

(
1
+
c
)
⩽

T
k

(
x
)
⩽
2

c

/

(
1
+
c
)
}

\{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\}

, where

c
c

is a positive parameter. The weight function of the polynomials of the second kind is
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace 1 minus left-parenthesis left-parenthesis 1 plus c right-parenthesis squared slash 4 normal c right-parenthesis upper T Subscript k Superscript 2 Baseline left-parenthesis x right-parenthesis right-brace Superscript 1 slash 2 Baseline slash StartAbsoluteValue upper U Subscript k minus 1 Baseline left-parenthesis x right-parenthesis EndAbsoluteValue">

{
1
−
(

(
1
+
c

)
2

/

4
c
)

T
k
2

(
x
)

}

1

/

2

/

|

U

k
−
1

(
x
)

|

{\{ 1 - ({(1 + c)^2}/4\operatorname {c} )T_k^2(x)\} ^{1/2}}/|{U_{k - 1}}(x)|

when

c
⩾
1

c \geqslant 1

. When

c
>
1

c > 1

we pick up discrete masses located at the zeros of

U

k
−
1

(
x
)

{U_{k - 1}}(x)

. The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their

q
q

-analogues are also treated. Their continuous spectra are also the above mentioned set. The

q
q

-analogues include a sieved version of the Rogers

q
q

-ultraspherical polynomials and another set of

q
q

-ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.
Classification of metabelian 𝑝-groups
Authors: Chou Wu Nan
Published: June 2010

Let

G
G

be a two-generator metabelian group of exponent

p
p

with class of nilpotence

c
c

, where

c
⩽
p
−
1

c \leqslant p - 1

and

p
p

is an odd prime. In this paper, we shall consider the classification problem when

|

G
2

/

G
3

|

=
p

|{G_2}/{G_3}| = p

,

|

G
3

/

G
4

|

=

p
2

|{G_3}/{G_4}| = {p^2}

and

|

G
4

/

G
5

|

⩽

p
2

|{G_4}/{G_5}| \leqslant {p^2}

.
Joint spectra and analytic set-valued functions
Authors: Klimek M.
Published: June 2010

We investigate analyticity of joint spectra of

A
m

{A^m}

-valued holomorphic mappings, where

A
A

denotes a complex Banach algebra. We show also that if

K
K

is an analytic set-valued function whose values are compact subsets of

C

n

{{\mathbf {C}}^n}

and

d
d

is the transfinite diameter in

C

n

{{\mathbf {C}}^n}

, then the upper-semicontinuous regularization of

log
⁡
d
(
K
)

\log d(K)

is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit’s Scarcity Theorem.
On 𝐾₃ of truncated polynomial rings
Authors: Aisbett Janet
Published: June 2010

Group homology spectral sequences are used to investigate

K
3

{K_3}

of truncated polynomial rings. If

F
F

is a finite field of odd characteristic, we show that relative

K
2

{K_2}

of the pair

(
F

[
t
]

/

(

t
q

)
,

(

t
k

)
)

(F\left [ t \right ]/({t^q}),\,({t^k}))

, which has been identified by van der Kallen and Stienstra, is isomorphic to

K
3

(
F

[
t
]

/

(

t
k

)
,

(
t
)
)

{K_3}(F\left [ t \right ]/({t^k}),\,(t))

when

q
q

is sufficiently large. We also show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 3 left-parenthesis SL bold upper Z left-bracket t right-bracket slash left-parenthesis t Superscript k Baseline right-parenthesis semicolon bold upper Z right-parenthesis equals bold upper Z Superscript k minus 1 Baseline circled-plus bold upper Z slash 24">

H
3

(

SL

Z

[
t
]

/

(

t
k

)
;

Z

)
=

Z

k
−
1

⊕

Z

/

24

{H_3}({\text {SL}}\,{\mathbf {Z}}\left [ t \right ]/({t^k});{\mathbf {Z}}) = {{\mathbf {Z}}^{k - 1}} \oplus {\mathbf {Z}}/24

and is isomorphic to the associated

K
3

{K_3}

group modulo an elementary abelian

2
2

-group.
𝐺-deformations and some generalizations of H. Weyl’s tube theorem
Authors: Kowalski Oldřich , Vanhecke Lieven
Published: June 2010

We prove an invariance theorem for the volumes of tubes about submanifolds in arbitrary analytic Riemannian manifolds under

G
G

-deformations of the second order. For locally symmetric spaces or two-point homogeneous spaces we give stronger invariance theorems using only

G
G

-deformations of the first order. All these results can be viewed as generalizations of the result of H. Weyl about isometric deformations and the volumes of tubes in spaces of constant curvature. They are derived from a new formula for the volume of a tube about a submanifold.
Some remarks on deformations of minimal surfaces
Authors: Rosenberg Harold , Toubiana Éric
Published: June 2010

We consider complete minimal surfaces (c.m.s.’s) in

R
3

{R^3}

and their deformations.

M
1

{M_1}

is an

ε
\varepsilon

-deformation of

M
0

{M_0}

if

M
1

{M_1}

is a graph over

M
0

{M_0}

in an

ε
\varepsilon

tubular neighborhood of

M
0

{M_0}

and

M
1

{M_1}

is

ε

C
1

\varepsilon \;{C^1}

-close to

M
0

{M_0}

. A minimal surface

M
M

is isolated if all c.m.s.’s which are sufficiently small deformations of

M
M

are congruent to

M
M

. In this paper we construct an example of a nonisolated c.m.s. It is modelled on a

4
4

-punctured sphere and is of finite total curvature. On the other hand, we prove that a c.m.s. discovered by Meeks and Jorge, modelled on the sphere punctured at the fourth roots of unity, is isolated.
An extremal problem for analytic functions with prescribed zeros and 𝑟th derivative in 𝐻^{∞}
Authors: Horwitz A. , Newman D. J.
Published: June 2010

Let

(

α
1

,
…
,

α
n

)

({\alpha _1}, \ldots ,{\alpha _n})

be

n
n

points in the unit disc

U
U

. Suppose

g
g

is analytic in

U
U

,

g
(

α
1

)
=
⋯
=
g
(

α
n

)
=
0

g({\alpha _1}) = \cdots = g({\alpha _n}) = 0

(multiplicities included), and

‖
g
′

‖

∞

≤
1

\|g\prime \|_{\infty } \leq 1

. Then we prove that

|

g
(
z
)

|

≤

|

ϕ
(
z
)

|

|g(z)| \leq |\phi (z)|

for all

z
∈
U

z \in U

, where

ϕ
(

α
1

)
=
⋯
=
ϕ
(

α
n

)
=
0

\phi ({\alpha _1}) = \cdots = \phi ({\alpha _n}) = 0

and

ϕ
′
(
z
)

\phi \prime (z)

is a Blaschke product of order

n
−
1

n - 1

. We extend this result in a natural way to convex domains

D
D

with analytic boundary. For

D
D

not convex we show that there is no extremal function

ϕ
\phi

.
𝐿^{𝑝} inequalities for stopping times of diffusions
Authors: DeBlassie R. Dante
Published: June 2010

Let

X
t

{X_t}

be a solution to a stochastic differential equation. Easily verified conditions on the coefficients of the equation give

L
p

{L^p}

inequalities for stopping times of

X
t

{X_t}

and the maximal function. An application to Brownian motion with radial drift is also discussed.
Automorphic images of commutative subspace lattices
Authors: Harrison K. J. , Longstaff W. E.
Published: June 2010

Let

C
(
H
)

C(H)

denote the lattice of all (closed) subspaces of a complex, separable Hilbert space

H
H

. Let

(

AC)

({\text {AC)}}

be the following condition that a subspace lattice

F

⊆
C
(
H
)

\mathcal {F} \subseteq C(H)

may or may not satisfy: (AC)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row script upper F equals phi left-parenthesis script upper L right-parenthesis for some lattice automorphism phi of upper C left-parenthesis upper H right-parenthesis 2nd Row and some commutative subspace lattice script upper L subset-of-or-equal-to upper C left-parenthesis upper H right-parenthesis period EndLayout">

F

=
ϕ
(

L

)

for

some

lattice

automorphism

ϕ

of

C
(
H
)

and

some

commutative

subspace

lattice

L

⊆
C
(
H
)
.

\begin {array}{*{20}{c}} {\mathcal {F} = \phi (\mathcal {L})\;{\text {for}}\;{\text {some}}\;{\text {lattice}}\;{\text {automorphism}}\;\phi \;{\text {of}}\;C(H)} \\ {{\text {and}}\;{\text {some}}\;{\text {commutative}}\;{\text {subspace}}\;{\text {lattice}}\;\mathcal {L} \subseteq C(H).} \\ \end {array}

\]
Then

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

A

⊆

B

\mathcal {A} \subseteq \mathcal {B}

for some Boolean algebra subspace lattice

B

⊆
C
(
H
)

\mathcal {B} \subseteq C(H)

with the property that, for every

K
,
L
∈

B

K,L \in \mathcal {B}

, the vector sum

K
+
L

K + L

is closed. If

F

\mathcal {F}

is finite, then

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

F

\mathcal {F}

is distributive and

K
+
L

K + L

is closed for every

K
,
L
∈

F

K,L \in \mathcal {F}

. In finite dimensions

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

F

\mathcal {F}

is distributive. Every

F

\mathcal {F}

satisfying

(

AC

)

({\text {AC}})

is reflexive. For such

F

\mathcal {F}

, given vectors

x
,
y
∈
H

x,y \in H

, the solvability of the equation

T
x
=
y

Tx = y

for

T
∈
Alg

F

T \in \operatorname {Alg}\,\mathcal {F}

is investigated.
Stable finitely homogeneous structures
Authors: Cherlin G. , Lachlan A. H.
Published: June 2010

Let

L
L

be a finite relational language and

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

denote the class of countable

L
L

-structures which are stable and homogeneous. The main result of the paper is that there exists a natural number

c
(
L
)

c(L)

such that for any transitive

M

∈
Hom
⁡
(
L
;
ω
)

\mathcal {M} \in \operatorname {Hom}(L;\omega )

, if

E
E

is a maximal

0
0

-definable equivalence relation on

M

\mathcal {M}

, then either

|

M

/

E

|

>
c
(
L
)

|\mathcal {M}/E| > c(L)

, or

M

/

E

\mathcal {M}/E

is coordinatizable. In an earlier paper the second author analyzed certain subclasses

Hom
⁡
(
L
,
r
)
 
(
r
>
ω
)

\operatorname {Hom}(L, r)\ (r > \omega )

of

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

for all sufficiently small

r
r

. Thus the earlier analysis now applies to

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

.
Isometries on 𝐿_{𝑝,1}
Authors: Carothers N. L. , Turett B.
Published: June 2010

The extreme points of the sphere of the Lorentz function space

L

p
,
1

[
0
,
1
]

{L_{p,1}}[0,1]

are computed. As an application, the linear isometries from

L

p
,
1

{L_{p,1}}

into itself are completely described.
Hypoelliptic convolution equations in the space 𝒦’ₑ
Authors: Pahk Dae Hyeon
Published: June 2010

We consider convolution equations in the space

K

e
′

\mathcal {K}_e’

of distributions which "grow" no faster than

exp
⁡
(

e

k

|

x

|

)

\exp ({e^{k|x|}})

for some constant

k
k

. Our main results are to find conditions for convolution operators to be hypoelliptic in

K

e
′

\mathcal {K}_e’

in terms of their Fourier transforms.
On the zeros of successive derivatives of even Laguerre-Pólya functions
Authors: Shen Li-Chien
Published: June 2010

Using "method of steepest descent", we prove that the final set (in the sense of Polya) of a class of even Laguerre-Polya functions is the entire real axis.
Tameness and local normal bases for objects of finite Hopf algebras
Authors: Childs Lindsay N. , Hurley Susan
Published: June 2010

LetRRbe a commutative ring,SSanRR-algebra,HHa HopfRRalgebra, both finitely generated and projective asRR-modules, and supposeSSis anHH-object, so thatH∗=HomR(H,R){H^{\ast }} = {\operatorname {Hom} _R}(H,R)acts onSSvia a measuring. LetIIbe the space of left integrals ofH∗{H^{\ast }}. We saySShas normal basis ifS≅HS \cong HasH∗{H^{\ast }}modules, andSShas local normal bases ifSp≅Hp{S_p} \cong {H_p}asHp∗H_p^{\ast }-modules for all prime idealsppofRR. WhenRRis a perfect field,HHis commutative and cocommutative, and certain obvious necessary conditions onSShold, thenSShas normal basis if and only ifIS=R=SH∗IS = R = {S^{{H^{\ast }}}}. IfRRis a domain with quotient fieldKK,HHis cocommutative, andL=S⊗RKL = S \otimes {}_RKhas normal basis as(H∗⊗K)({H^{\ast }} \otimes K)-module, thenSShas local normal bases if and only ifIS=R=SH∗IS = R = {S^{{H^{\ast }}}}.
Isoperimetric inequalities for the least harmonic majorant of |𝑥|^{𝑝}
Authors: Sakai Makoto
Published: June 2010

Let

D
D

be an open set in the

d
d

-dimensional Euclidean space

R

d

{{\mathbf {R}}^d}

containing the origin

0
0

and let

h

(
p
)

(
x
,
D
)

{h^{(p)}}(x,D)

be the least harmonic majorant of

|

x

|

p

|x{|^p}

in

D
D

, where

0
>
p
>
∞

0 > p > \infty

if

d
⩾
2

d \geqslant 2

and

1
⩽
p
>
∞

1 \leqslant p > \infty

if

d
=
1

d = 1

. We shall be concerned with the following isoperimetric inequalities

h

(
p
)

(
0
,
D

)

1

/

p

⩽
c
r
(
D
)

{h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)

, where

r
(
D
)

r(D)

denotes the volume radius of

D
D

, namely, a ball with radius

r
(
D
)

r(D)

has the same volume as

D
D

has and

c
c

is a constant dependent on

d
d

and

p
p

but independent of

D
D

. We fix

d
d

and denote by

c
(
p
)

c(p)

the infimum of such constants

c
c

. As a function of

p
p

,

c
(
p
)

c(p)

is nondecreasing and satisfies

c
(
p
)
⩾
1

c(p) \geqslant 1

. We shall show (1) there are positive constants

C
1

{C_1}

and

C
2

{C_2}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 1 p Superscript left-parenthesis d minus 1 right-parenthesis slash d Baseline less-than-or-slanted-equals c left-parenthesis p right-parenthesis less-than-or-slanted-equals upper C 2 p Superscript left-parenthesis d minus 1 right-parenthesis slash d">

C
1

p

(
d
−
1
)

/

d

⩽
c
(
p
)
⩽

C
2

p

(
d
−
1
)

/

d

{C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}

for

p
⩾
1

p \geqslant 1

, (2)

c
(
p
)
=
1

c(p) = 1

if

p
⩽
d
+

2

1
−
d

p \leqslant d + {2^{1 - d}}

. Many estimations of

h

(
p
)

(
0
,
D
)

{h^{(p)}}(0,D)

and their applications are also given.
The asymptotic behavior near the crest of waves of extreme form
Authors: McLeod J. B.
Published: June 2010

The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle

2
3

π

\frac {2} {3}\pi

. Amick and Fraenkel have investigated the asymptotic behavior of the free surface near the crest and obtained an asymptotic expansion for this behavior, but are unable to say whether the leading term in this expansion has a nonzero coefficient (and so whether it is in fact the leading term or not). The present paper shows that the coefficient is nonzero and determines its sign.
A dimension formula for Hermitian modular cusp forms of degree two
Authors: Eie Min King
Published: June 2010

An explicit dimension formula for the vector space of Hermitian modular cusp forms of degree two with respect to the modular group
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma 2 left-parenthesis bold upper Z left-bracket i right-bracket right-parenthesis equals upper S upper U left-parenthesis 2 comma 2 right-parenthesis intersection upper M 4 left-parenthesis bold upper Z left-bracket i right-bracket right-parenthesis">

Γ
2

(

Z

[
i
]
)
=
SU
⁡
(
2
,
2
)
∩

M
4

(

Z

[
i
]
)

{\Gamma _2}({\mathbf {Z}}[i]) = \operatorname {SU} (2,2) \cap {M_4}({\mathbf {Z}}[i])

is obtained via the Selberg trace formula and its arithmetic properties. Also, a generating function for the graded ring of Hermitian cusp forms of degree two is given.
Complex geometry and the asymptotics of Harish-Chandra modules for real reductive Lie groups. I
Authors: Casian Luis G. , Collingwood David H.
Published: June 2010

Let

G
G

be a connected semisimple real matrix group. It is now apparent that the representation theory of

G
G

is intimately connected with the complex geometry of the flag variety

B

\mathcal {B}

. By studying appropriate orbit structures on

B

\mathcal {B}

, we are naturally led to representation theory in the category of Harish-Chandra modules

H

C

\mathcal {H}\mathcal {C}

, or the representation theory of category

O

′

\mathcal {O}’

. The Jacquet functor

J
:

H

C

→

O

′

J:\mathcal {H}\mathcal {C} \to \mathcal {O}’

has proved a useful tool in converting "

H

C

\mathcal {H}\mathcal {C}

problems" into "

O

′

\mathcal {O}’

problems," which are often more tractable. In this paper, we advance the philosophy that the complex geometry of

B

\mathcal {B}

, associated to

H

C

\mathcal {H}\mathcal {C}

and

O

′

\mathcal {O}’

, interacts in a natural way with the functor

J
J

, leading to deep new information on the structure of Jacquet modules. This, in turn, gives new insight into the structure of certain nilpotent cohomology groups associated to Harish-Chandra modules. Our techniques are based upon many of the ideas present in the proof of the Kazhdan-Lusztig conjectures and Bernstein’s proof of the Jantzen conjecture.
Hypergeometric functions over finite fields
Authors: Greene John
Published: June 2010

In this paper the analogy between the character sum expansion of a complex-valued function over

GF

(
p
)

{\text {GF}}(p)

and the power series expansion of an analytic function is exploited in order to develop an analogue for hypergeometric series over finite fields. It is shown that such functions satisfy many summation and transformation formulas analogous to their classical counterparts.
A cohomological pairing of half-forms
Authors: Robinson P. L.
Published: June 2010

Blattner and Rawnsley have constructed half-forms for regular polarizations of arbitrary index. We show how to pair these half-forms into a line bundle fashioned purely from the symplectic data, with no assumption on the intersection of the polarizations. Our pairing agrees with the regular BKS pairing when the polarizations are positive.
Isotype submodules of 𝑝-local balanced projective groups
Authors: Lane Mark
Published: June 2010

By giving necessary and sufficient conditions for two isotype submodules of a

p
p

-local balanced projective group to be equivalent, we are able to introduce a general theory of isotype submodules of

p
p

-local balanced projective groups (or

I
B

IB

modules). Numerous applications of the above result are available particularly for the special class of

I
B

IB

modules introduced by Wick (known as SKT modules). We first show that the class of SKT modules is closed under direct summands, and then we are able to show that if

H
H

appears as an isotype submodule of the

p
p

-local balanced projective group

G
G

such that

G

/

H

G/H

is the coproduct of countably generated torsion groups, then

H
H

is an SKT module. Finally we show that

I
B

IB

modules satisfy general structural properties such as transitivity, full transitivity, and the equivalence of

p
α

{p^\alpha }

-high submodules.
Hamiltonian analysis of the generalized problem of Bolza
Authors: Clarke F. H.
Published: June 2010

On étudie le problème généralisé de Bolza en calcul des variations. Presented at the International Conference on the Calculus of Variations held to honour the memory of Leonida Tonelli, Scuola Normale Superiore, Pisa, March 1986. On obtient des conditions nécessaires en forme hamiltonienne, sous des hypothèses moins exigeantes qu’antérieurement, en particulier sans qualification sur les contraintes. Le lien avec les problèmes de contrôle optimal est développé, ainsi que l’apport de ces conditions à la théorie de la régularité de la solution. We obtain necessary conditions in Hamiltonian form for the generalized problem of Bolza in the calculus of variations. These are proven in part by an extension to Hamiltonians of Tonelli’s method of auxiliary Lagrangians. One version of the conditions is of a new character since it is obtained in the absence of any constraint qualification on the data. A new regularity theorem is shown to be a consequence of the necessary conditions.
Scalar curvature functions in a conformal class of metrics and conformal transformations
Authors: Bourguignon Jean-Pierre , Ezin Jean-Pierre
Published: June 2010

This article addresses the problem of prescribing the scalar curvature in a conformal class. (For the standard conformal class on the

2
2

-sphere, this is usually referred to as the Nirenberg problem.) Thanks to the action of the conformal group, integrability conditions due to J. L. Kazdan and F. W. Warner are extended, and shown to be universal. A counterexample to a conjecture by J. L. Kazdan on the role of first spherical harmonics in these integrability conditions on the standard sphere is given. Using the action of the conformal groups, some existence results are also given.
The Fraser-Horn and Apple properties
Authors: Berman Joel , Blok W. J.
Published: June 2010

We consider varietiesV\mathcal {V}in which finite direct products are skew-free and in which the congruence lattices of finite directly indecomposables have a unique coatom. We associate withV\mathcal {V}a family of derived varieties,d(V)d(\mathcal {V}): a variety ind(V)d(\mathcal {V})is generated by algebrasA{\mathbf {A}}where the universe ofA{\mathbf {A}}consists of a congruence class of the coatomic congruence of a finite directly indecomposable algebraB∈V{\mathbf {B}} \in \mathcal {V}and the operations ofA{\mathbf {A}}are those ofB{\mathbf {B}}that preserve this congruence class. We also consider the prime variety ofV\mathcal {V}, denotedV0{\mathcal {V}_0}, generated by all finite simple algebras inV\mathcal {V}. We show how the structure of finite algebras inV\mathcal {V}is determined to a considerable extent byV0{\mathcal {V}_0}andd(V)d(\mathcal {V}). In particular, the freeV\mathcal {V}-algebra onnngenerators,FV(n){{\mathbf {F}}_\mathcal {V}}(n), has as many directly indecomposable factors asFV0(n){{\mathbf {F}}_{{\mathcal {V}_0}}}(n)and the structure of these factors is determined by the varietiesd(V)d(\mathcal {V}). This allows us to produce in many cases explicit formulas for the cardinality ofFV(n){{\mathbf {F}}_\mathcal {V}}(n). Our work generalizes the structure theory of discriminator varieties and, more generally, that of arithmetical semisimple varieties. The paper contains many examples of algebraic systems that have been investigated in different contexts; we show how these all fit into a general scheme.
The topology of resolution towers
Authors: Akbulut Selman , King Henry
Published: June 2010

An obstruction theory is given to determine when a space has a resolution tower. This can be used to decide whether or not the space is homeomorphic to a real algebraic set.
Strong multiplicity theorems for 𝐺𝐿(𝑛)
Authors: Gilbert George T.
Published: June 2010

Let

π
=
⊗

π
υ

\pi = \otimes {\pi _\upsilon }

be a cuspidal automorphic representation of

G
L
(
n
,

F
A

)

GL(n,{F_A})

, where

F
A

{F_A}

denotes the adeles of a number field

F
F

. Let

E
E

be a Galois extension of

F
F

and let

{
g
}

\{ g\}

denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components

π
υ

{\pi _\upsilon }

whenever the Frobenius of the prime

υ
\upsilon

is

{
g
}

\{ g\}

, showing that such representations are often easily described and finite in number. This generalizes a result of Moreno [Bull. Amer. Math. Soc. 11 (1984), pp. 180-182].
The normal subgroup structure of the Picard group
Authors: Fine Benjamin , Newman Morris
Published: June 2010

The Picard group

Γ
\Gamma

is

P
S

L
2

(
Z
[
i
]
)

PS{L_2}(Z[i])

, the group of linear fractional transformations with Gaussian integer coefficients. We examine the structure of the normal subgroups of

Γ
\Gamma

. In particular we give a complete classification of the normal subgroups for indices less than

60
60

and show that beyond this there are large gaps in the possible indices. This classification depends on the structure of the derived series. Finally we give examples of normal noncongruence subgroups.
Where does the 𝐿^{𝑝}-norm of a weighted polynomial live?
Authors: Mhaskar H. N. , Saff E. B.
Published: June 2010

For a general class of nonnegative weight functions

w
(
x
)

w(x)

having bounded or unbounded support

Σ
⊂

R

\Sigma \subset {\mathbf {R}}

, the authors have previously characterized the smallest compact set

S

w

{\mathfrak {S}_w}

, having the property that for every

n
=
1
,

2
,
…

n = 1,\,2, \ldots

and every polynomial

P
P

of degree

⩽
n

\leqslant n

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue left-bracket w left-parenthesis x right-parenthesis right-bracket Superscript n Baseline upper P left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis normal upper Sigma right-parenthesis Baseline equals StartAbsoluteValue EndAbsoluteValue left-bracket w left-parenthesis x right-parenthesis right-bracket Superscript n Baseline upper P left-parenthesis x right-parenthesis StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis German upper S Sub Subscript w Subscript right-parenthesis">

|

|

[
w
(
x
)

]
n

P
(
x
)

|

|

L
∞

(
Σ
)

=

|

|

[
w
(
x
)

]
n

P
(
x
)

|

|

L
∞

(

S

w

)

||{[w(x)]^n}P(x)|{|_{{L^\infty }(\Sigma )}} = ||{[w(x)]^n}P(x)|{|_{{L^\infty }({\mathfrak {S}_w})}}

\]
. In the present paper we prove that, under mild conditions on

w
w

, the

L
p

{L^p}

-norms

(
0
>
p
>
∞
)

(0 > p > \infty )

of such weighted polynomials also "live" on

S

w

{\mathfrak {S}_w}

in the sense that for each

η
>
0

\eta > 0

there exist a compact set

Δ
\Delta

with Lebesgue measure

m
(
Δ
)
>
η

m(\Delta ) > \eta

and positive constants

c
1

{c_1}

,

c
2

{c_2}

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue EndAbsoluteValue w Superscript n Baseline upper P StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis normal upper Sigma right-parenthesis Baseline less-than-or-slanted-equals left-parenthesis 1 plus c 1 exp left-parenthesis minus c 2 n right-parenthesis right-parenthesis StartAbsoluteValue EndAbsoluteValue w Superscript n Baseline upper P StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript p Subscript left-parenthesis German upper S Sub Subscript w Subscript union normal upper Delta right-parenthesis">

|

|

w
n

P

|

|

L
p

(
Σ
)

⩽
(
1
+

c
1

exp
⁡
(
−

c
2

n
)
)

|

|

w
n

P

|

|

L
p

(

S

w

∪
Δ
)

||{w^n}P|{|_{{L^p}(\Sigma )}} \leqslant (1 + {c_1}\exp ( - {c_2}n))||{w^n}P|{|_{{L^p}({\mathfrak {S}_w} \cup \Delta )}}

\]
. As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.