bluuebunny/crossref_metadata_2025 huggingface dataset:
Sobolev inequalities for weight spaces and supercontractivity
Authors: Rosen Jay
Published: July 2010

For

ϕ
∈

C
2

(

R

n

)

\phi \in {C^2}({{\mathbf {R}}^n})

with

ϕ
(
x
)
=
a

|

x

|

1
+
s

\phi (x) = a|x{|^{1 + s}}

for

|

x

|

⩾

x
0

,
a
,
s
>
0

|x| \geqslant {x_0},a,s > 0

, define the measure

d
μ
=
exp
⁡
(
−
2
ϕ
)

d
n

x

d\mu = \exp ( - 2\phi ){d^n}x

on

R

n

{{\mathbf {R}}^n}

. We show that for any

k
∈

Z

+

k \in {{\mathbf {Z}}^ + }

\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row integral StartAbsoluteValue f EndAbsoluteValue squared StartAbsoluteValue log base 10 left-parenthesis StartAbsoluteValue f EndAbsoluteValue right-parenthesis EndAbsoluteValue Superscript 2 s k slash left-parenthesis s plus 1 right-parenthesis Baseline d mu 2nd Row less-than-or-slanted-equals c left-brace sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue equals 0 Overscript k Endscripts double-vertical-bar upper D Superscript alpha Baseline f double-vertical-bar Subscript upper L 2 left-parenthesis d mu right-parenthesis Superscript 2 Baseline plus double-vertical-bar f double-vertical-bar Subscript upper L 2 left-parenthesis d mu right-parenthesis Superscript 2 Baseline bullet StartAbsoluteValue log base 10 left-parenthesis double-vertical-bar f double-vertical-bar Subscript upper L 2 left-parenthesis d mu right-parenthesis Baseline right-parenthesis EndAbsoluteValue Superscript 2 s k slash left-parenthesis s plus 1 right-parenthesis Baseline right-brace EndLayout">

∫

|

f

|

2

|

lg
⁡
(

|

f

|

)

|

2
s
k

/

(
s
+
1
)

d
μ

⩽
c

{

∑

|

α

|

=
0

k

‖

D
α

f
‖

L
2

(
d
μ
)

2

+

‖
f
‖

L
2

(
d
μ
)

2

∙

|

lg
⁡
(

‖
f
‖

L
2

(
d
μ
)

)

|

2
s
k

/

(
s
+
1
)

}

\begin {array}{*{20}{c}} {\int {|f{|^2}|\lg (|f|){|^{2sk/(s + 1)}}d\mu } } \hfill \\ { \leqslant c\left \{ {\sum \limits _{|\alpha | = 0}^k {\left \|{D^\alpha }f\right \|_{{L_2}(d\mu )}^2 + \left \|f\right \|_{{L_2}(d\mu )}^2 \bullet |\lg (\left \|f\right \|_{{L_2}(d\mu )}){|^{2sk/(s + 1)}}} } \right \}} \hfill \\ \end {array}

\]
As a consequence we prove
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="e Superscript minus t nabla Super Superscript asterisk Superscript dot nabla Baseline colon upper L Subscript q Baseline left-parenthesis bold upper R Superscript n Baseline comma d mu right-parenthesis right-arrow upper L Subscript p Baseline left-parenthesis bold upper R Superscript n Baseline comma d mu right-parenthesis comma p comma q not-equals 1 comma normal infinity">

e

−
t

∇
∗

⋅
∇

:

L
q

(

R

n

,
d
μ
)
→

L
p

(

R

n

,
d
μ
)
,
p
,
q
≠
1
,
∞

{e^{ - t{\nabla ^\ast } \cdot \nabla }}:{L_q}({{\mathbf {R}}^n},d\mu ) \to {L_p}({{\mathbf {R}}^n},d\mu ),p,q \ne 1,\infty

, is bounded for all

t
>
0

t > 0

.
On the local stability of differential forms
Authors: Golubitsky Martin , Tischler David
Published: July 2010

In this paper we determine which germs of differential s-forms on an n-manifold are stable (in the sense of Martinet). We show that when

s
≠
1

s \ne 1

or when

s
=
1

s = 1

and

n
⩽
4

n \leqslant 4

Martinet had found almost all of the possible examples. The most interesting result states that for certain generic singularities of 1-forms on 4-manifolds an infinite dimensional moduli space occurs in the classification of the 1-forms with this given singularity type up to equivalence by pull-back via a diffeomorphism.
The Remez exchange algorithm for approximation with linear restrictions
Authors: Chalmers Bruce L.
Published: July 2010

This paper demonstrates a Remez exchange algorithm applicable to approximation of real-valued continuous functions of a real variable by polynomials of degree smaller than n with various linear restrictions. As special cases are included the notion of restricted derivatives approximation (examples of which are monotone and convex approximation and restricted range approximation) and the notion of approximation with restrictions at poised Birkhoff data (examples of which are bounded coefficients approximation,

ε
\varepsilon

-interpolator approximation, and polynomial approximation with restrictions at Hermite and “Ferguson-Atkinson-Sharma” data and pyramid matrix data). Furthermore the exchange procedure is completely simplified in all the cases of approximation with restrictions at poised Birkhoff data. Also results are obtained in the cases of general linear restrictions where the Haar condition prevails. In the other cases (e.g., monotone approximation) the exchange in general requires essentially a matrix inversion, although insight into the exchange is provided and partial alternation results are obtained which lead to simplifications.
On bounded univalent functions whose ranges contain a fixed disk
Authors: Barnard Roger W.
Published: July 2010

Let

S

\mathcal {S}

denote the standard normalized class of regular, univalent functions in

K
=

K
1

=
{
z
:

|

z

|

>
1
}

K = {K_1} = \{ z:|z| > 1\}

. Let

F

\mathcal {F}

be a given compact subclass of

S

\mathcal {S}

. We consider the following two problems. Problem 1. Find

max

|

a
2

|

\max |{a_2}|

for

f
∈

F

f \in \mathcal {F}

. Problem 2. For

|

z

|

=
r
>
1

|z| = r > 1

, find the

max

(
min
)

|

f
(
z
)

|

\max \;(\min )|f(z)|

for

f
∈

F

f \in \mathcal {F}

. In this paper we are concerned with the subclass
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper S Subscript d Superscript asterisk Baseline left-parenthesis upper M right-parenthesis equals StartSet f element-of script upper S colon upper K Subscript d Baseline subset-of f left-parenthesis upper K right-parenthesis subset-of upper K Subscript upper M Baseline EndSet">

S

d
∗

(
M
)
=
{
f
∈

S

:

K
d

⊂
f
(
K
)
⊂

K
M

}

\mathcal {S}_d^\ast (M) = \{ f \in \mathcal {S}:{K_d} \subset f(K) \subset {K_M}\}

. Through the use of the Julia variational formula and the Loewner theory we determine the extremal functions for Problems 1 and 2 for the class

S

d
∗

(
M
)

\mathcal {S}_d^\ast (M)

, for all d, M such that

1
4

⩽
d
⩽
1
⩽
M
⩽
∞

\tfrac {1}{4} \leqslant d \leqslant 1 \leqslant M \leqslant \infty

.
Hausdorff measure functions in the space of compact subsets of the unit interval
Authors: Goodey P. R.
Published: July 2010

The work done in this paper is the result of an attempt to classify those functions h for which the corresponding Hausdorff measure of

F

[
0
,
1
]

\mathcal {F}[0,1]

is zero. A partial characterization is achieved and in doing this some problems of E. Boardman are solved.
Embedding Stieltjes-Volterra integral equations in Stieltjes integral equations
Authors: Gibson William L.
Published: July 2010

J. A. Reneke has shown that the linear Stieltjes-Volterra integral equations studied by D. B. Hinton can be transformed into Stieltjes integral equations of the type studied by J. S. Mac Nerney. By taking advantage of the nonlinear nature of Mac Nerney’s results, Reneke was able to extend Hinton’s existence theorem to a nonlinear setting. In this paper, we use Reneke’s embedding technique to generalize several other of Hinton’s results, and we characterize completely, in the linear case, the range of Reneke’s embedding transformation.
Extensions of Haar measure to relatively large nonmeasurable subgroups
Authors: Peterson H. Leroy
Published: July 2010

Let G be a locally compact group, with

λ
\lambda

a left Haar measure on G. A subgroup is large if it has finite index; a relatively large subgroup of G is a large subgroup of an open subgroup. In §1 we have an existence theorem for relatively large nonopen subgroups, and we observe that such subgroups are not

λ
\lambda

-measurable. This motivates the development, in §2, of a left-invariant countably additive extension

λ
+

{\lambda ^ + }

of

λ
\lambda

, which includes in its domain all unions of left translates of a given relatively large subgroup K. For an arbitrarily chosen family

K

I

{\mathcal {K}_I}

of relatively large subgroups of G, we define (in §3) a finitely additive measure

λ
I
+

\lambda _I^ +

such that, for any

K
∈

K

I

,

λ
I
+

K \in {\mathcal {K}_I},\lambda _I^ +

is an extension of the corresponding

λ
+

{\lambda ^ + }

defined in §2. An example shows that

λ
I
+

\lambda _I^ +

need not be countably additive. Finally, in §4, we observe some aspects of the relationship between

λ
+

{\lambda ^ + }

-measurable and

λ
\lambda

-measurable functions, in the context of existing literature on extensions of Haar measure. In particular, we generalize the well-known proposition that

λ
\lambda

-measurable characters are continuous.
The independence ratio and genus of a graph
Authors: Albertson Michael O. , Hutchinson Joan P.
Published: July 2010

In this paper we study the relationship between the genus of a graph and the ratio of the independence number to the number of vertices.
Cohomological dimension of a group with respect to finite modules
Authors: Martínez Juan José
Published: July 2010

The purpose of this paper is to compare the cohomological dimension of a group, relative to finite modules, with the cohomological dimension, in the usual sense, of its profinite completion. The basic tool used to perform this comparison is certain stable cohomology of the group. The reason is that there exists a spectral sequence which relates the continuous cohomology of the profinite completion, with coefficients in this stable cohomology, to the ordinary cohomology of the group. Moreover, the direct method of connecting the cohomology of the group with the profinite cohomology of its completion arises from the edge effects on the base of this spectral sequence.
Cone bundles
Authors: McCrory Clint
Published: July 2010

A theory of normal bundles for locally knotted codimension two embeddings of PL manifolds is developed. The classifying space for this theory is Cappell and Shaneson’s space

B
R

N
2

BR{N_2}

.
Lifting idempotents and exchange rings
Authors: Nicholson W. K.
Published: July 2010

Idempotents can be lifted modulo a one-sided ideal L of a ring R if, given

x
∈
R

x \in R

with

x
−

x
2

∈
L

x - {x^2} \in L

, there exists an idempotent

e
∈
R

e \in R

such that

e
−
x
∈
L

e - x \in L

. Rings in which idempotents can be lifted modulo every left (equivalently right) ideal are studied and are shown to coincide with the exchange rings of Warfield. Some results of Warfield are deduced and it is shown that a projective module P has the finite exchange property if and only if, whenever

P
=
N
+
M

P = N + M

where N and M are submodules, there is a decomposition

P
=
A
⊕
B

P = A \oplus B

with

A
⊆
N

A \subseteq N

and

B
⊆
M

B \subseteq M

.
Two-descent for elliptic curves in characteristic two
Authors: Kramer Kenneth
Published: July 2010

This paper is a study of two-descent to find an upper bound for the rank of the Mordell-Weil group

A
(
F
)

A(F)

of an elliptic curve A defined over a field F of characteristic two. It includes local and global duality theorems which are the analogs of known results for descent by an isogeny whose degree is relatively prime to the characteristic of the field of definition.
Plugging flows
Authors: Percell Peter B. , Wilson F. Wesley
Published: July 2010

A plug construction is a local modification of a nonsingular flow which severs certain kinds of recurrence properties. In this paper we investigate the effect of plug constructions on minimal sets, the nonwandering set, and the chain recurrent set and the explosions of these sets when a plug construction is perturbed.
On a notion of smallness for subsets of the Baire space
Authors: Kechris Alexander S.
Published: July 2010

Let us call a set

A
⊆

ω
ω

A \subseteq {\omega ^\omega }

of functions from

ω
\omega

into

ω

σ

\omega \;\sigma

-bounded if there is a countable sequence of functions

{

α
n

:
n
∈
ω
}
⊆

ω
ω

\{ {\alpha _n}:n \in \omega \} \subseteq {\omega ^\omega }

such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of

ω
ω

{\omega ^\omega }

. We show that most of the usual definability results about the structure of countable subsets of

ω
ω

{\omega ^\omega }

have corresponding versions which hold about

σ
\sigma

-bounded subsets of

ω
ω

{\omega ^\omega }

. For example, we show that every

Σ

2
n
+
1

1

σ

\Sigma _{2n + 1}^1\;\sigma

-bounded subset of

ω
ω

{\omega ^\omega }

has a

Δ

2
n
+
1

1

\Delta _{2n + 1}^1

“bound”

{

α
m

:
m
∈
ω
}

\{ {\alpha _m}:m \in \omega \}

and also that for any

n
⩾
0

n \geqslant 0

there are largest

σ
\sigma

-bounded

Π

2
n
+
1

1

\Pi _{2n + 1}^1

and

Σ

2
n
+
2

1

\Sigma _{2n + 2}^1

sets. We need here the axiom of projective determinacy if

n
⩾
1

n \geqslant 1

. In order to study the notion of

σ
\sigma

-boundedness a simple game is devised which plays here a role similar to that of the standard

∗

^\ast

-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the

∗

^\ast

- and

∗
∗

^{ \ast \ast }

- (or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of

ω
ω

{\omega ^\omega }

whose special cases include countability, being of the first category and

σ
\sigma

-boundedness and for which one can generalize all the main results of the present paper.
PL involutions of fibered 3-manifolds
Authors: Kim Paik Kee , Tollefson Jeffrey L.
Published: July 2010

Let h be a PL involution of

F
×
[
0
,
1
]

F \times [0,1]

such that

h
(
F
×
{
0
,
1
}
)
=
F
×
{
0
,
1
}

h(F \times \{ 0,1\} ) = F \times \{ 0,1\}

, where F is a compact 2-manifold. It is shown that h is equivalent to an involution

h
′

h’

of the form

h
′

(
x
,
t
)
=
(
g
(
x
)
,
λ
(
t
)
)

h’(x,t) = (g(x),\lambda (t))

. This result is applied to classify the PL involutions of closed, orientable, Seifert manifolds when the fixed-point set contains a two-dimensional component of negative characteristic.
Deformations of Lie subgroups
Authors: Coppersmith Don
Published: July 2010

We give rigidity and universality theorems for embedded deformations of Lie subgroups. If

K
⊂
H
⊂
G

K \subset H \subset G

are Lie groups, with

H
1

(
K
,
g

/

h
)
=
0

{H^1}(K,g/h) = 0

, then for every

C
∞

{C^\infty }

deformation of H, a conjugate of K lies in each nearby fiber

H
s

{H_s}

. If

H
⊂
G

H \subset G

with

H
2

(
H
,
g

/

h
)
=
0

{H^2}(H,g/h) = 0

, then there is a universal “weak” analytic deformation of H, whose base space is a manifold with tangent plane canonically identified with

Ker
⁡

δ
1

\operatorname {Ker} {\delta ^1}

.
The structure of generalized Morse minimal sets on 𝑛 symbols
Authors: Martin John C.
Published: July 2010

A class of bisequences on n symbols is constructed which includes the generalized Morse sequences introduced by Keane. Those which give rise to strictly ergodic sets are characterized, and the spectrum of the shift operator on these systems is investigated. It is shown that in certain cases the shift operator has partly discrete and partly continuous spectrum. The theorems generalize results of Keane on generalized Morse sequences and a theorem of Kakutani regarding a particular strictly transitive sequence on four symbols. Another special case yields information on the spectrum of certain substitution minimal sets.
Weak 𝐿₁ characterizations of Poisson integrals, Green potentials and 𝐻^{𝑝} spaces
Authors: Sjögren Peter
Published: July 2010

Our main result can be described as follows. A subharmonic function u in a suitable domain

Ω
\Omega

in

R

n

{{\mathbf {R}}^n}

is the difference of a Poisson integral and a Green potential if and only if u divided by the distance to

∂
Ω

\partial \Omega

is in weak

L
1

{L_1}

in

Ω
\Omega

. Similar conditions are given for a harmonic function to be the Poisson integral of an

L
p

{L_p}

function on

∂
Ω

\partial \Omega

. Iterated Poisson integrals in a polydisc are also considered. As corollaries, we get weak

L
1

{L_1}

characterizations of

H
p

{H^p}

spaces of different kinds.
Symplectic Stiefel harmonics and holomorphic representations of symplectic groups
Authors: Ton-That Tuong
Published: July 2010

Let

I
k

{I_k}

denote the identity matrix of order k and set
\[

s
k

=

[

0

a
m
p
;

−

I
k

I
k

a
m
p
;
0

]

.

{s_k} = \left [ {\begin {array}{*{20}{c}} 0 & { - {I_k}} \\ {{I_k}} & 0 \\ \end {array} } \right ].

\]
Let

Sp

(
k
,

C

)

{\text {Sp}}(k,{\mathbf {C}})

denote the group of all complex

2
k
×
k

2k \times k

matrices which satisfy the equation

g

s
k

g
t

=

s
k

g{s_k}{g^t} = {s_k}

. Let E be the linear space of all

n
×
2
k

n \times 2k

complex matrices with

k
⩾
n

k \geqslant n

, and let

S
(

E
∗

)

S({E^\ast })

denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of

Sp

(
k
,

C

)

{\text {Sp}}(k,{\mathbf {C}})

, which is obtained by right translation on

S
(

E
∗

)

S({E^\ast })

, leads to a concrete and simple realization of all irreducible holomorphic representations of

Sp

(
k
,

C

)

{\text {Sp}}(k,{\mathbf {C}})

. In connection with this realization, a theory of symplectic Stiefel harmonics is also established. This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold.
A singular semilinear equation in 𝐿¹(𝑅)
Authors: Crandall Michael G. , Evans Lawrence C.
Published: July 2010

Let

β
\beta

be a positive and nondecreasing function on R. The boundary-value problem

β
(
u
)
−
u
=
f
,

u
′

(
±
∞
)
=
0

\beta (u) - u = f,u’( \pm \infty ) = 0

is considered for

f
∈

L
1

(

R

)

f \in {L^1}({\mathbf {R}})

. It is shown that this problem can have a solution only if

β
\beta

is integrable near

−
∞

- \infty

, and that if this is the case, then the problem has a solution exactly when

∫

−
∞

∞

f
(
x
)
d
x
>
0

\smallint _{ - \infty }^\infty f(x)dx > 0

.
Embedding of closed categories into monoidal closed categories
Authors: Laplaza Miguel L.
Published: July 2010

S. Eilenberg and G. M. Kelly have defined a closed category as a category with internal homomorphism functor, left Yoneda natural arrows, unity object and suitable coherence axioms. A monoidal closed category is a closed category with an associative tensor product which is adjoint to the int-hom. This paper proves that a closed category can be embedded in a monoidal closed category: the embedding preserves any associative tensor product which may exist. Besides the usual tools of the theory of closed categories the proof uses the results of B. Day on promonoidal structures.
The generalized Green’s function for an 𝑛th order linear differential operator
Authors: Locker John
Published: July 2010

The generalized Green’s function

K
(
t
,
s
)

K(t,s)

for an nth order linear differential operator L is characterized in terms of the 2nth order differential operators

L

L
∗

L{L^\ast }

and

L
∗

L

{L^\ast }L

. The development is operator oriented and takes place in the Hilbert space

L
2

[
a
,
b
]

{L^2}[a,b]

. Two features of the characterization are a determination of the jumps occurring in the derivatives of orders n,

n
+
1
,
…
,
2
n
−
1

n + 1, \ldots ,2n - 1

at

t
=
s

t = s

and a determination of the boundary conditions satisfied by the functions

K
(
a
,
⋅
)

K(a, \cdot )

and

K
(
b
,
⋅
)

K(b,\cdot )

. Several examples are given to illustrate the properties of the generalized Green’s function.
A general extremal problem for the class of close-to-convex functions
Authors: Milcetich John G.
Published: July 2010

Forβ⩾0,Kβ\beta \geqslant 0,{K_\beta }denotes the set of functionsf(z)=z+a2z2+⋯f(z) = z + {a_2}{z^2} + \cdotsdefined on the unit discUwith the representationf′(z)=apβ(z)s(z)/zf’(z) = a{p^\beta }(z)s(z)/z, wherea∈Ca \in C,pis an analytic function with positive real part inU, andsis a normalized starlike function. If0⩽β⩽10 \leqslant \beta \leqslant 1, andζ∈U\zeta \in U, letF(u,v)F(u,v)be analytic in a neighborhood of{(f(ζ),ζ):f∈Kβ}\{ (f(\zeta ),\zeta ):f \in {K_\beta }\}. Thenmax{Re⁡F(f(ζ),ζ):f∈Kβ}\max \{ \operatorname {Re} F(f(\zeta ),\zeta ):f \in {K_\beta }\}occurs for a function of the form\[<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis equals left-parenthesis beta plus 1 right-parenthesis Superscript negative 1 Baseline left-parenthesis x minus y right-parenthesis Superscript negative 1 Baseline left-bracket left-parenthesis 1 plus x z right-parenthesis Superscript beta plus 1 Baseline left-parenthesis 1 plus y z right-parenthesis Superscript negative beta minus 1 Baseline minus 1 right-bracket comma">f(z)=(β+1)−1(x−y)−1[(1+xz)β+1(1+yz)−β−1−1],f(z) = {(\beta + 1)^{ - 1}}{(x - y)^{ - 1}}[{(1 + xz)^{\beta + 1}}{(1 + yz)^{ - \beta - 1}} - 1],\]where|x|=|y|=1|x| = |y| = 1andx≠yx \ne y. If0>β>10 > \beta > 1these are the only extremal functions. A consequence of this result is the determination of the value region{f(ζ)/ζ:f∈Kβ}\{ f(\zeta )/\zeta :f \in {K_\beta }\}as<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace left-parenthesis beta plus 1 right-parenthesis Superscript negative 1 Baseline left-parenthesis s minus t right-parenthesis Superscript negative 1 Baseline left-bracket left-parenthesis 1 plus s right-parenthesis Superscript beta plus 1 Baseline left-parenthesis 1 plus t right-parenthesis Superscript negative beta minus 1 Baseline minus 1 right-bracket colon StartAbsoluteValue s EndAbsoluteValue comma StartAbsoluteValue t EndAbsoluteValue less-than-or-slanted-equals StartAbsoluteValue zeta EndAbsoluteValue right-brace">{(β+1)−1(s−t)−1[(1+s)β+1(1+t)−β−1−1]:|s|,|t|⩽|ζ|}\{ {(\beta + 1)^{ - 1}}{(s - t)^{ - 1}}[{(1 + s)^{\beta + 1}}{(1 + t)^{ - \beta - 1}} - 1]:|s|,|t| \leqslant |\zeta |\}.
Variations, characteristic classes, and the obstruction to mapping smooth to continuous cohomology
Authors: Mostow Mark A.
Published: July 2010

In a recent paper, the author gave an example of a singular foliation on

R

2

{{\mathbf {R}}^2}

for which it is impossible to map the de Rham cohomology

T

DR

{T_{{\text {DR}}}}

to the continuous singular cohomology

T

c

{T_{\text {c}}}

(in the sense of Bott and Haefliger’s continuous cohomology of spaces with two topologies) compatibly with evaluation of cohomology classes on homology classes. In this paper the obstruction to mapping

T

DR

{T_{{\text {DR}}}}

to

T

c

{T_{\text {c}}}

is pinpointed by defining a whole family of cohomology theories

T

k
,
m
,
n

{T_{k,m,n}}

, based on cochains which vary in a

C
k

{C^k}

manner, which mediate between the two. It is shown that the obstruction vanishes on nonsingularly foliated manifolds. The cohomology theories are extended to Haefliger’s classifying space

(
B

Γ
q

→
B

J
q

)

(B{\Gamma _q} \to B{J_q})

, with its germ and jet topologies, by using a notion of differentiable space similar to those of J. W. Smith and K. T. Chen. The author proposes that certain of the

T

k
m
n

{T_{kmn}}

be used instead of

T

c

{T_{\text {c}}}

to study Bott and Haefliger’s conjecture that the continuous cohomology of

(
B

Γ
q

→
B

J
q

)

(B{\Gamma _q} \to B{J_q})

equals the relative Gel’fand-Fuks cohomology

H
∗

(

a

q

,

O
q

)

{H^\ast }({\mathfrak {a}_q},{O_q})

. It is shown that

T

k
m
n

(
B

Γ
q

→
B

J
q

)

{T_{kmn}}(B{\Gamma _q} \to B{J_q})

may contain new characteristic classes for foliations which vary only in a

C
k

{C^k}

manner when a foliation is varied smoothly.
Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors
Authors: Cushman R.
Published: July 2010

Any analytic symplectic diffeomorphism

Φ
\Phi

of a symplectic manifold M is the Poincaré map of a real analytic Hamiltonian vector field

X
H

{X_H}

. If

Φ
\Phi

does not have an analytic integral, then

X
H

{X_H}

has no analytic integral which is not a power series in H. Let

M
=

R

2

M = {{\mathbf {R}}^2}

. If

Φ
\Phi

has a finite contact homoclinic point, then

Φ
\Phi

is nonintegrable. Also Moser’s polynomial mapping is nonintegrable.
Weak Chebyshev subspaces and continuous selections for the metric projection
Authors: Nürnberger Günther , Sommer Manfred
Published: July 2010

Let G be an n-dimensional subspace of

C
[
a
,
b
]

C[a,b]

. It is shown that there exists a continuous selection for the metric projection if for each f in

C
[
a
,
b
]

C[a,b]

there exists exactly one alternation element

g
f

{g_f}

, i.e., a best approximation for f such that for some

a
⩽

x
0

>
⋯
>

x
n

⩽
b

a \leqslant {x_0} > \cdots > {x_n} \leqslant b

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon left-parenthesis negative 1 right-parenthesis Superscript i Baseline left-parenthesis f minus g Subscript f Baseline right-parenthesis left-parenthesis x Subscript i Baseline right-parenthesis equals double-vertical-bar f minus g Subscript f Baseline double-vertical-bar comma i equals 0 comma ellipsis comma n comma epsilon equals plus-or-minus 1 period">

ε

(
−
1

)
i

(
f
−

g
f

)
(

x
i

)
=

‖

f
−

g
f

‖

,

i
=
0
,
…
,
n
,
ε
=
±
1.

\varepsilon {( - 1)^i}(f - {g_f})({x_i}) = \left \| {f - {g_f}} \right \|,\quad i = 0, \ldots ,n,\varepsilon = \pm 1.

\]
Further it is shown that this condition is fulfilled if and only if G is a weak Chebyshev subspace with the property that each g in G,

g
≠
0

g \ne 0

, has at most n distinct zeros. These results generalize in a certain sense results of Lazar, Morris and Wulbert for

n
=
1

n = 1

and Brown for

n
=
5

n = 5

.
Maximum principles, gradient estimates, and weak solutions for second-order partial differential equations
Authors: Bertiger William
Published: July 2010

Weak solutions to second order elliptic equations and the first derivatives of these solutions are shown to satisfy

L
p

{L^p}

bounds. Classical second order equations with nonnegative characteristic form are also considered. It is proved that auxiliary functions of the gradient of a solution must satisfy a maximum principle. This result is extended to higher order derivatives and systems.
Sampling theorems for nonstationary random processes
Authors: Lee Alan J.
Published: July 2010

Consider a second order stochastic process

{
X
(
t
)
,
t
∈

R

}

\{ X(t),t \in \textbf {R}\}

, and let

H
(
X
)

H(X)

be the Hilbert space generated by the random variables of the process. The process is said to be linearly determined by its samples

{
X
(
n
h
)
,
n
∈

Z

}

\{ X(nh),n \in \textbf {Z}\}

if the random variables

X
(
n
h
)

X(nh)

generate

H
(
X
)

H(X)

. In this paper we give a sufficient condition for a wide class of nonstationary processes to be determined by their samples, and present sampling theorems for such processes. We also consider similar problems for harmonizable processes indexed by LCA groups having suitable subgroups.
The metabelian 𝑝-groups of maximal class
Authors: Miech R. J.
Published: July 2010

This paper presents a solution to the isomorphism problem for the set of metabelian p-groups of maximal class.
Strong differentiability of Lipschitz functions
Authors: Neugebauer C. J.
Published: July 2010

Let F be a differentiation basis in

R
n

{R^n}

, i.e., a family of measurable sets S contracting to 0 such that

‖

M
F

f

‖

p

⩽

A
p

‖
f
‖

p

{\left \| {{M_F}f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p}

, where

M
F

{M_F}

is the Hardy-Littlewood maximal operator. For

f
∈

Λ
α

p
q

f \in \Lambda _\alpha ^{pq}

, we let

E
F

(
f
)

{E_F}(f)

be the complement of the Lebesgue set of f relative to F, and we show that

E
F

{E_F}

has

L
α

p
q

L_\alpha ^{pq}

-capacity 0, where

L
α

p
q

L_\alpha ^{pq}

is a capacity associated with

Λ
α

p
q

\Lambda _\alpha ^{pq}

in much the same way as the Bessel capacity

B

α
p

{B_{\alpha p}}

is associated with

L
α
p

L_\alpha ^p

.
Semimodular functions and combinatorial geometries
Authors: Nguyen Hien Quang
Published: July 2010

A point-lattice

L

\mathfrak {L}

being given, to any normalized, nondecreasing, integer-valued, semimodular function f defined on

L

\mathfrak {L}

, we can associate a class of combinatorial geometries called expansions of f. The family of expansions of f is shown to have a largest element for the weak map order,

E
(
f
)

E(f)

, the free expansion of f. Expansions generalize and clarify the relationship between two known constructions, one defined by R. P. Dilworth, the other by J. Edmonds and G.-C. Rota. Further applications are developed for solving two extremal problems of semimodular functions: characterizing (1) extremal rays of the convex cone of real-valued, nondecreasing, semimodular functions defined on a finite set; (2) combinatorial geometries which are extremal for the decomposition into a sum.
Systems of 𝑛 partial differential equations in 𝑛 unknown functions: the conjecture of M. Janet
Authors: Johnson Joseph
Published: July 2010

It was conjectured by Janet that an analytic solution to a system of n “independent” analytic differential equations in n unknown functions if not isolated must depend on at least one unknown function of

m
−
1

m - 1

variables plus possibly other functions of fewer than m variables. Here m is the dimension of the complex domain on which the equations and the solution are given. An algebraic generalization of the linear form of the conjecture is proven. Also the result is extended to give a nonlinear version.
Topologies on spaces of vector-valued continuous functions
Authors: Khurana Surjit Singh
Published: July 2010

Topologies

β
0

,

β
1

,
β
,

β
∞

,

β

∞
c

{\beta _0},{\beta _1},\beta ,{\beta _\infty },{\beta _{\infty c}}

are defined on

C
b

(
X
,
E
)

{C_b}(X,E)

, the space of all bounded, continuous functions from a completely regular Hausdorff space X, into E, a normed space, and their duals are determined. Also many properties of these topologies are proved.
Curves with large tangent space
Authors: Becker Joseph , Gurjar Rajendra
Published: July 2010

Theorem. Let V be a complex analytic variety irreducible at a point

p
∈
V

p \in V

. Givén any integer l, there exists an analytic curve

C
l

{C_l}

on V passing through p and irreducible at p such that the germs of

C
l

{C_l}

and V at p are isomorphic up to order l.
Existence, stability, and compactness in the 𝛼-norm for partial functional differential equations
Authors: Travis C. C. , Webb G. F.
Published: July 2010

The abstract ordinary functional differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a slash d t right-parenthesis u left-parenthesis t right-parenthesis equals minus upper A u left-parenthesis t right-parenthesis plus upper F left-parenthesis u Subscript t Baseline right-parenthesis comma u 0 equals phi">

(
a

/

d
t
)
u
(
t
)
=
−
A
u
(
t
)
+
F
(

u
t

)
,

u
0

=
ϕ

(a/dt)u(t) = - Au(t) + F({u_t}),{u_0} = \phi

, is studied, where

−
A

- A

is the infinitesimal generator of an analytic semigroup of linear operators and F is continuous with respect to a fractional power of A.
The minimum norm projection on 𝐶²-manifolds in 𝑅ⁿ
Authors: Abatzoglou Theagenis J.
Published: July 2010

We study the notion of best approximation from a point

x
∈

R
n

x \in {R^n}

to a

C
2

{C^2}

-manifold. Using the concept of radius of curvature, introduced by J. R. Rice, we obtain a formula for the Fréchet derivative of the minimum norm projection (best approximation) of

x
∈

R
n

x \in {R^n}

into the manifold. We also compute the norm of this derivative in terms of the radius of curvature.
A note on the operator 𝑋→𝐴𝑋-𝑋𝐵
Authors: Fialkow L.
Published: July 2010

If A and B are bounded linear operators on an infinite dimensional complex Hilbert space

H

\mathcal {H}

, let

τ
(
X
)

=

A
X

−

X
B

\tau (X)\, = \,AX\, - \,XB

(X in

L

(

H

)

\mathcal {L}(\mathcal {H})

). It is proved that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis tau right-parenthesis equals sigma left-parenthesis tau vertical-bar upper C Subscript p Baseline right-parenthesis left-parenthesis 1 less-than-or-slanted-equals p less-than-or-slanted-equals normal infinity right-parenthesis">

σ
(
τ
)

=

σ
(
τ

|

C
p

)

(
1

⩽

p

⩽
∞
)

\sigma (\tau )\, = \,\sigma (\tau |{C_p})\,(1\, \leqslant \,p\, \leqslant \infty )

, where, for

1

⩽
p

>

∞

1\, \leqslant p\, > \,\infty

,

C
p

{C_p}

is the Schatten p-ideal, and

C
∞

{C_\infty }

is the ideal of all compact operators in

L

(

H

)

\mathcal {L}(\mathcal {H})

. Analogues of this result for the parts of the spectrum are obtained and sufficient conditions are given for

τ
\tau

to be injective. It is also proved that if A and B are quasisimilar, then the right essential spectrum of A intersects the left essential spectrum of B.
Interpolation by complex splines
Authors: Tzimbalario J.
Published: July 2010

In this paper we solve the problem of interpolation by certain class of cardinal complex splines. This solution is used to complete the study of cardinal trigonometric splines started in [10] and also to give shorter proofs and to complete the results found for the interpolation problem by complex splines over the unit circle by I. J. Schoenberg [9], J. H. Ahlberg, E. N. Nilson and J. L. Walsh [1].
Analytically decomposable operators
Authors: Lange Ridgley
Published: July 2010

The author introduces the notion of an analytically decomposable operator which generalizes the decomposable operator due to C. Foias in that the spectral decompositions of the underlying Banach space (1) admit a wider class of invariant subspaces called “analytically invariant” and (2) span the space only densely. It is shown that analytic decomposability is stable under the functional calculus, direct sums and restrictions to certain kinds of invariant subspaces, as well as perturbation by commuting scalar operators. It is fundamental for many of these results that every analytically decomposable operator has the single-valued extension property. An extensive investigation of analytically invariant subspaces is given. The author shows by example that this class is distinct from those of spectral maximal and hyperinvariant subspaces, but he further shows that analytically invariant subspaces have many useful spectral properties. Some applications of the general theory are made. For example, it is shown that under certain restrictions an analytically decomposable operator is decomposable.
Approximation theorems for uniformly continuous functions
Authors: Hager Anthony W.
Published: July 2010

Let X be a set, A a family of real-valued functions on X which contains the constants,

μ
A

{\mu _A}

the weak uniformity generated by A, and

U
(

μ
A

X
)

U({\mu _A}X)

the collection of uniformly continuous functions to the real line R. The problem is how to construct

U
(

μ
A

X
)

U({\mu _A}X)

from A. The main result here is: For A a vector lattice, the collection of suprema of countable, finitely A-equiuniform, order-one subsets of

A
+

{A^ + }

is uniformly dense in

U
(

μ
A

X
)

U({\mu _A}X)

. Two less technical corollaries: If A is a vector lattice (resp., vector space), then the collection of functions which are finitely A-uniform and uniformly locally-A (resp., uniformly locally piecewise-A) is uniformly dense in

U
(

μ
A

X
)

U({\mu _A}X)

. Further, for any A, a finitely A-uniform function is just a composition

F
∘
(

a
1

,
…
,

a
p

)

F \circ ({a_1}, \ldots ,{a_p})

for some

a
1

,
…
,

a
p

∈
A

{a_1}, \ldots ,{a_p} \in A

and F uniformly continuous on the range of

(

a
1

,
…
,

a
p

)

({a_1}, \ldots ,{a_p})

in

R
p

{R^p}

. Thus, such compositions are dense in

U
(

μ
A

X
)

U({\mu _A}X)

. For

B
U
(

μ
A

X
)

BU({\mu _A}X)

, the compositions with

F
∈
B
U
(

R
p

)

F \in BU({R^p})

are dense (B denoting bounded functions). So, in a sense, to know

U
(

μ
A

X
)

U({\mu _A}X)

it suffices to know A and subspaces of the spaces

R
p

{R^p}

, and to know

B
U
(

μ
A

X
)

BU({\mu _A}X)

, A and the spaces

R
p

{R^p}

suffice.
Stability of isometries
Authors: Gruber Peter M.
Published: July 2010

A map

T
:
E
→
F

T:E \to F

(E, F Banach spaces) is called an

ε
\varepsilon

-isometry if
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue double-vertical-bar upper T left-parenthesis x right-parenthesis minus upper T left-parenthesis y right-parenthesis double-vertical-bar minus double-vertical-bar x minus y double-vertical-bar EndAbsoluteValue less-than-or-slanted-equals epsilon">

|

‖

T
(
x
)
−
T
(
y
)

‖

−

‖

x
−
y

‖

|

⩽
ε

\left |\,{\left \| {T(x)-T(y)} \right \|-\left \|{x -y}\right \|}\,\right |\, \leqslant \varepsilon

whenever

x
,

y
∈
E

x,\,y \in E

. Hyers and Ulam raised the problem whether there exists a constant

κ
\kappa

, depending only on E, F, such that for every surjective

ε
\varepsilon

-isometry

T
:
E
→
F

T:E \to F

there exists an isometry

I
:
E
→
F

I:E \to F

with

‖

T
(
x
)
−
I
(
x
)

‖

⩽
κ
ε

{\left \| {T(x) - I(x)} \right \|}\leqslant \kappa \varepsilon

for every

x
∈
E

x \in E

. It is shown that, whenever this problem has a solution for E, F, one can assume

κ
⩽
5

\kappa \leqslant 5

. In particular this holds true in the finite dimensional case.
Semifree actions on finite groups on homotopy spheres
Authors: Ewing John
Published: July 2010

We show that for any finite group the group of semifree actions on homotopy spheres of some fixed even dimension is finite, provided that the dimension of the fixed point set is greater than 2. The argument shows that for such an action the normal bundle to the fixed point set is equivariantly, stably trivial.
Group actions on 𝐴_{𝑘}-manifolds
Authors: Ku Hsü Tung , Ku Mei Chin
Published: July 2010

By an

A
k

{A_k}

-manifold we mean a connected manifold with elements

w
i

∈

H
1

(
M
)
,

1
⩽
i
⩽
k

{w_i} \in {H^1}(M),\,1 \leqslant i \leqslant k

, such that

w
1

∪

⋯
∪

w
k

≠
0

{w_1} \cup \, \cdots \cup \,{w_k} \ne 0

. In this paper we study the fixed point set, degree of symmetry, semisimple degree of symmetry and gaps of transformation groups on

A
k

{A_k}

-manifolds.
The eigenvalue spectrum as moduli for flat tori
Authors: Wolpert Scott
Published: July 2010

A flat torus T carries a natural Laplace Beltrami operator. It is a conjecture that the spectrum of the Laplace Beltrami operator determines T modulo isometries. We prove that, with the exception of a subvariety in the moduli space of flat tori, this conjecture is true. A description of the subvariety is given.
Paracompactness of box products of compact spaces
Authors: Kunen Kenneth
Published: July 2010

We consider box products of countably many compact Hausdorff spaces. Under the continuum hypothesis, the product is paracompact iff its Lindelöf degree is no more than the continuum; in particular, the product is paracompact if each space has weight continuum or less, or if each space is dispersed. Some partial results are proved under Martin’s axiom.
Persistent manifolds are normally hyperbolic
Authors: Mañé Ricardo
Published: July 2010

Let M be a smooth manifold,

f
:

M

∣
a

C

1

f:\,M\,\mid \text {a}\,{{C}^{1}}

diffeomorphism and

V
⊂
M

a

C

1

V \subset M\,{\text {a}}\,{{\text {C}}^1}

compact submanifold without boundary invariant under f (i.e.

f

(
V
)

=

V

f\left ( V \right )\, = \,V

). We say that V is a persistent manifold for f if there exists a compact neighborhood U of V such that

∩

n

∈

z

f
n

(
U
)

=

V

{ \cap _{n\, \in \,{\textbf {z}}}}\,{f^n}\left ( U \right )\, = \,V

, and for all diffeomorphisms

g
:

M

∣

g:\,M\,\mid

near to f in the

C
1

{C^1}

topology the set

V
g

=

∩

n

∈

z

g
n

(
U
)

{V_g}\, = \,{ \cap _{n\, \in \,{\textbf {z}}}}{g^n}\left ( U \right )

is a

C
1

{C^1}

submanifold without boundary

C
1

{C^1}

near to V. Several authors studied sufficient conditions for persistence of invariant manifolds. Hirsch, Pugh and Shub proved that normally hyperbolic manifolds are persistent, where normally hyperbolic means that there exist a Tf-invariant splitting

T
M

/

V

=

N
s

V

⊕

N
u

V

⊕

T
V

TM/V\, = \,{N^s}V\, \oplus \,{N^u}V\, \oplus \,TV

and constants

K

>

0

K\, > \,0

,

0

>

λ

>

1

0\, > \,\lambda \, > \,1

such that:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row double-vertical-bar left-parenthesis upper T f right-parenthesis Superscript n Baseline slash upper N Subscript x Superscript s Baseline upper V double-vertical-bar less-than-or-equal-to upper K lamda Superscript n Baseline comma double-vertical-bar left-parenthesis upper T f right-parenthesis Superscript negative n Baseline slash upper N Subscript x Superscript u Baseline upper V double-vertical-bar less-than-or-equal-to upper K lamda Superscript n Baseline comma double-vertical-bar left-parenthesis upper T f right-parenthesis Superscript n Baseline slash upper N Subscript x Superscript s Baseline upper V double-vertical-bar dot double-vertical-bar left-parenthesis upper T f right-parenthesis Superscript negative n Baseline slash upper T Subscript f Sub Superscript n Subscript left-parenthesis x right-parenthesis Baseline upper V double-vertical-bar less-than-or-equal-to upper K lamda Superscript n EndLayout">

‖

(

T
f

)

n

/

N
x
s

V

‖

≤

K

λ
n

,

‖

(

T
f

)

−
n

/

N
x
u

V

‖

≤

K

λ
n

,

‖

(

T
f

)

n

/

N
x
s

V

‖

⋅

‖

(

T
f

)

−
n

/

T

f
n

(
x
)

V

‖

≤

K

λ
n

\begin {gathered} \left \| {{{\left ( {Tf} \right )}^n}/N_x^sV} \right \|\, \leq \,K{\lambda ^n},\,\left \| {{{\left ( {Tf} \right )}^{ - n}}/N_x^uV} \right \|\, \leq \,K{\lambda ^n}, \left \| {{{\left ( {Tf} \right )}^n}/N_x^sV} \right \|\, \cdot \,\left \| {{{\left ( {Tf} \right )}^{ - n}}/{T_{{f^n}\left ( x \right )}}V} \right \|\, \leq \,K{\lambda ^n} \end {gathered}

\]
for all

n

>

0

n\, > \,0

,

x

∈

V

x\, \in \,V

. In this paper we prove the converse result, namely that persistent manifolds are normally hyperbolic.
Intertwining differential operators for 𝑀𝑝(𝑛,𝑅) and 𝑆𝑈(𝑛,𝑛)
Authors: Jakobsen Hans Plesner
Published: July 2010

For each of the two series of groups, three series of representations

U
n

{U_n}

,

D
n

{D_n}

, and

H
n

(
n
∈
Z
)

{H_n}(n \in Z)

are considered. For each series of representations there is a differential operator with the property, that raised to the nth power

(
n
>
0
)

(n > 0)

, it intertwines the representations indexed by

−
n

- n

and n. The operators are generalizations of the d’Alembertian, the Diracoperator and a combination of the two. Unitarity of subquotients of representations indexed by negative integers is derived from the intertwining relations.
Invariance of the 𝐿-regularity of compact sets in 𝐶^{𝑁} under holomorphic mappings
Authors: Pleśniak W.
Published: July 2010

The property for a polynomially convex compact set E in

C
N

{C^N}

that the Siciak extremal function

Φ
E

{\Phi _E}

be continuous or, equivalently, that E satisfy some Bernstein type inequality, is proved to be invariant under a large class of holomorphic mappings with values in

C
M

(
M
⩽
N
)

{C^M}(M \leqslant N)

including all open holomorphic mappings. Local specifications of this result are also given.
Partitions and sums and products of integers
Authors: Hindman Neil
Published: July 2010

The principal result of the paper is that, if

r

>

ω

r\, > \,\omega

and

{

A
i

}

i
>
r

{\{ {A_i}\} _{i > r}}

is a partition of

ω
\omega

, then there exist

i

>

r

i\, > \,r

and infinite subsets B and C of

ω
\omega

such that

∑
F

∈

A
i

\sum F\, \in \,{A_i}

and

∏

G

∈

A
i

\prod {G\, \in \,{A_i}}

whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.
Starlike, convex, close-to-convex, spiral-like, and Φ-like maps in a commutative Banach algebra with identity
Authors: Heath L. F. , Suffridge T. J.
Published: July 2010

Let

C
(
X
)

C(X)

be the space of continuous functions on a compact

T
2

{T_2}

-space

X
X

where each point of

X
X

is a

G
δ

{G_\delta }

. If

F
:
B
→
C
(
X
)

F:B \to C(X)

is a biholomorphic (in the sense that

F
F

and

F

−
1

{F^{ - 1}}

are Fréchet differentiable) map of

B
=
{
f

|

‖
f
‖

>
1
}

B = \{ f\left | {\left \| f \right \|} \right . > 1\}

onto a convex domain with

D
F
(
0
)
=
I

DF(0) = I

, then

F
F

is Lorch analytic (i.e.,

D
F
(
f
)
(
g
)
=

a
f

g

DF(f)(g) = {a_f}g

for some

a

f

∈
C
(
X
)
)

{a_{f}} \in C(X))

). Let

R
R

be a commutative Banach algebra with identity such that the Gelfand homomorphism of

R
R

into

C
(

M

)

C(\mathcal {M})

is an isometry. Starlike, convex, close-to-convex, spirallike and

Φ
\Phi

-like functions are defined in

B
=
{
x
∈
R

|

‖
x
‖

>
1
}

B = \{ x \in R\left | {\left \| x \right \|} \right . > 1\}

for

L
L

-analytic functions in

B
B

and they are related to associated complex-valued holomorphic functions in

Δ
=
{
z
∈

C

|

|
z
|

>
1
}

\Delta = \{ z \in \left . \mathbf {C} \right |\left | z \right | > 1\}

.
Control problems governed by a pseudo-parabolic partial differential equation
Authors: White Luther W.
Published: July 2010

Let G be a bounded domain in

R
n

{R^n}

and

Q

=

G

×

(

0
,

T

)

Q\, = \,G\, \times \,\left ( {0,\,T} \right )

. We consider the solution

y

(
u
)

y\left ( u \right )

of the pseudo-parabolic initial-value problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row left-parenthesis 1 plus upper M left-parenthesis x right-parenthesis right-parenthesis y Subscript t Baseline left-parenthesis u right-parenthesis plus upper L left-parenthesis x right-parenthesis y left-parenthesis u right-parenthesis equals u in upper L squared left-parenthesis upper Q right-parenthesis comma 2nd Row y left-parenthesis dot comma 0 semicolon u right-parenthesis equals 0 in upper L squared left-parenthesis upper G right-parenthesis comma EndLayout">

(

1

+

M

(
x
)

)

y
t

(
u
)

+

L

(
x
)

y

(
u
)

=

u

in

L
2

(
Q
)

,

y

(

⋅
,

0
;

u

)

=

0

in

L
2

(
G
)

,

\begin{multline} \left ( {1\, + \,M\left ( x \right )} \right )\,{y_t}\,\left ( u \right )\, + \,L\left ( x \right )\,y\left ( u \right )\, = \,u\,{\text {in}}\,{L^2}\,\left ( Q \right ), \hfill \\ y\left ( { \cdot ,\,0;\,u} \right )\, = \,0\,{\text {in}}\,{L^2}\,\left ( G \right ), \hfill \\ \end{multline}

, to be the state corresponding to the control u. Here

M

(
x
)

M\left ( x \right )

and

L

(
x
)

L\left ( x \right )

are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control

u
0

{u_0}

in a fixed ball in

L
2

(
Q
)

{L^2}\left ( Q \right )

such that (i) the endpoint of the corresponding state

y

(

⋅
,

T
;

u
0

)

y\left ( { \cdot ,\,T;\,{u_0}} \right )

lies in a given neighborhood of a target Z in

L
2

(
G
)

{L^2}\left ( G \right )

and (ii)

u
0

{u_0}

minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.
Regular points of Lipschitz functions
Authors: Ioffe Alexander D.
Published: July 2010

Let f be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for f relative to S, and give a sufficient condition for a point

z

∈

S

z \, \in \, S

to be regular. This condition is then expressed in the particular case when f is

C
1

{C^1}

, and is used to obtain a new proof of Hoffman’s inequality in linear programming.