bluuebunny/crossref_metadata_2025 huggingface dataset:
Fine structure of the space of spherical minimal immersions
Authors: Gauchman Hillel , Toth Gabor
Published: July 2002

The space of congruence classes of full spherical minimal immersions

f
:

S
m

→

S
n

f:S^m\to S^n

of a given source dimension

m
m

and algebraic degree

p
p

is a compact convex body

M

m
p

\mathcal {M}_m^p

in a representation space

F

m
p

\mathcal {F}_m^p

of the special orthogonal group

S
O
(
m
+
1
)

SO(m+1)

. In Ann. of Math. 93 (1971), 43–62 DoCarmo and Wallach gave a lower bound for

F

m
p

\mathcal {F}_m^p

and conjectured that the estimate was sharp. Toth resolved this “exact dimension conjecture” positively so that all irreducible components of

F

m
p

\mathcal {F}_m^p

became known. The purpose of the present paper is to characterize each irreducible component

V
V

of

F

m
p

\mathcal {F}_m^p

in terms of the spherical minimal immersions represented by the slice

V
∩

M

m
p

V\cap \mathcal {M}_m^p

. Using this geometric insight, the recent examples of DeTurck and Ziller are located within

M

m
p

\mathcal {M}_m^p

.
Homology and some homotopy decompositions for the James filtration on spheres
Authors: Selick Paul
Published: July 2002

The filtrations on the James construction on spheres,

J

k

(

S

2
n

)

J_{k}\left (S^{2n}\right )

, have played a major role in the study of the double suspension

S

2
n
−
1

→

Ω
2

S

2
n
+
1

S^{2n-1}\to \Omega ^2 S^{2n+1}

and have been used to get information about the homotopy groups of spheres and Moore spaces and to construct product decompositions of related spaces. In this paper we calculate
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript asterisk Baseline left-parenthesis normal upper Omega upper J Subscript k Baseline left-parenthesis upper S Superscript 2 n Baseline right-parenthesis semicolon double-struck upper Z slash p double-struck upper Z right-parenthesis">

H
∗

(
Ω

J

k

(

S

2
n

)

;

Z

/

p

Z

)

H_*\left ( \Omega J_{k}\left (S^{2n}\right ); {\mathbb {Z}}/p{\mathbb {Z}}\right )

for odd primes 

p
p

. When

k
k

has the form

p
t

−
1

p^t-1

, the result is well known, but these are exceptional cases in which the homology has polynomial growth. We find that in general the homology has exponential growth and in some cases also has higher

p
p

-torsion. The calculations are applied to construct a

p
p

-local product decomposition of

Ω

J

k

(

S

2
n

)

\Omega J_{k}\left (S^{2n}\right )

for

k
>

p
2

−
p

k>p^2-p

which demonstrates a mod 

p
p

homotopy exponent in these cases.
The automorphism group of a coded system
Authors: Fiebig Doris , Fiebig Ulf-Rainer
Published: July 2002

We give a general construction of coded systems with an automorphism group isomorphic to

Z

⊕
G

\mathbf {Z}\oplus G

where

G
G

is any preassigned group which has a “continuous block presentation” (the isomorphism will map the shift to

(
1
,

e
G

)
)

(1,e_G))

. Several applications are given. In particular, we obtain automorphism groups of coded systems which are abelian, which are finitely generated and one which contains

Z

[
1

/

2
]

\mathbf {Z}[1/2]

. We show that any group which occurs as a subgroup of the automorphism group of some subshift with periodic points dense already occurs for some synchronized system.
Krull-Schmidt theorems in dimension 1
Authors: Levy Lawrence , Odenthal Charles
Published: July 2002

Let

Λ
\Lambda

be a semiprime, module-finite algebra over a commutative noetherian ring

R
R

of Krull dimension 1. We find necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated

Λ
\Lambda

-modules, and necessary and sufficient conditions for the Krull-Schmidt theorem to hold for all finitely generated torsionfree

Λ
\Lambda

-modules (called “

Λ
\Lambda

-lattices” in integral representation theory, and “maximal Cohen-Macaulay modules” in the dimension-one situation in commutative algebra).
Ω-inverse limit stability theorem
Authors: Ikeda Hiroshi
Published: July 2002

We prove that if an endomorphism

f
f

satisfies weak Axiom A and the no-cycles condition then

f
f

is

Ω
\Omega

-inverse limit stable. This result is a generalization of Smale’s

Ω
\Omega

-stability theorem from diffeomorphisms to endomorphisms.
Special values of symmetric hypergeometric functions
Authors: Baldassarri Francesco
Published: July 2002

We discuss thepp-adic formula (0.3) of P. Th. Young, in the framework of Dwork’s theory of the hypergeometric equation. We show that it gives the value at 0 of the Frobenius automorphism of the unit root subcrystal of the hypergeometric crystal. The unit disk at 0 is in fact singular for the differential equation under consideration, so that it’s not a priori clear that the Frobenius structure should extend to that disk. But the singularity islogarithmic, and it extends to a divisor with normal crossings relative toZp\mathbf {Z}_{p}inPZp1\mathbf {P}^{1}_{\mathbf {Z}_{p}}. We show that whenever the unit root subcrystal of the hypergeometric system has generically rank 1, it actually extends as alogarithmicFF-subcrystalto the unit disk at 0. So, in these optics, “singular classes are not supersingular”. If, in particular, the holomorphic solution at 0 isbounded, the extended logarithmicFF-crystal has no singualrity in the residue class of 0, so that it is anFF-crystal in the usual sense and the Frobenius operation is holomorphic. We examine in detail its analytic form.
An algebraic approach to multiparameter spectral theory
Authors: Grunenfelder Luzius , Kosir Tomaz
Published: July 2002

Root subspaces for multiprameter eigenvalue problems are described using coalgebraic techniques. An algorithm is given to construct bases for the root subspaces.
Functorial structure of units in a tensor product
Authors: Jaffe David
Published: July 2002

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let

k
k

be an algebraically closed field. Let

A
A

and

B
B

be reduced rings containing

k
k

, having connected spectra. Let

u
∈
A

⊗
k

B

u\in A\otimes _k\,B

be a unit. Then

u
=
a
⊗
b

u=a\otimes b

for some units

a
∈
A

a\in A

and

b
∈
B

b\in B

.

Here is a deeper consequence, stated for simplicity in the affine case only. Let

k
k

be a field, and let

φ
:
R
→
S

\varphi :R\to S

be a homomorphism of finitely generated

k
k

-algebras such that

Spec
⁡
(
φ
)

\operatorname {Spec}(\varphi )

is dominant. Assume that every irreducible component of

Spec
⁡
(

R

red

)

\operatorname {Spec}(R_{\operatorname {red}})

or

Spec
⁡
(

S

red

)

\operatorname {Spec}(S_{\operatorname {red}})

is geometrically integral and has a rational point. Let

B
→
C

B\to C

be a faithfully flat homomorphism of reduced

k
k

-algebras. For

A
A

a

k
k

-algebra, define

Q
(
A
)

Q(A)

to be

(
S

⊗
k

A

)
∗

/

(
R

⊗
k

A

)
∗

(S\otimes _k\,A)^*/(R\otimes _k\,A)^*

. Then

Q
Q

satisfies the following sheaf property: the sequence
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 right-arrow upper Q left-parenthesis upper B right-parenthesis right-arrow upper Q left-parenthesis upper C right-parenthesis right-arrow upper Q left-parenthesis upper C circled-times Subscript upper B Baseline upper C right-parenthesis">

0
→
Q
(
B
)
→
Q
(
C
)
→
Q
(
C

⊗
B

C
)

0\to Q(B)\to Q(C)\to Q(C\otimes _B\,C)

\]
is exact. This and another result are used to prove (5.2) of [R. Guralnick, D. B. Jaffe, W. Raskind and R. Wiegand, On the Picard group: torsion and the kernel induced by a faithfully flat map, J. of Algebra (to appear)].
Relatively free invariant algebras of finite reflection groups
Authors: Domokos Mátyás
Published: July 2002

Let

G
G

be a finite subgroup of

G

l

n

(
K
)

Gl_{n}(K)

(
K

(K

is a field of characteristic

0
0

and

n
≥
2
)

n\geq 2)

acting by linear substitution on a relatively free algebra

K
⟨

x

1

,
…
,

x

n

⟩

/

I

K\langle x_{1},\dots ,x_{n}\rangle /I

of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if

G
G

is a pseudo-reflection group and

I
I

contains the polynomial

[
[

x

2

,

x

1

]
,

x

1

]
.

[[x_{2},x_{1}],x_{1}].
Harish-Chandra’s Plancherel theorem for 𝔭-adic groups
Authors: Silberger Allan
Published: July 2002

Let

G
G

be a reductive

p

\mathfrak {p}

-adic group. In his paper, “The Plancherel Formula for Reductive

p

\mathfrak {p}

-adic Groups", Harish-Chandra summarized the theory underlying the Plancherel formula for

G
G

and sketched a proof of the Plancherel theorem for

G
G

. One step in the proof, stated as Theorem 11 in Harish-Chandra’s paper, has seemed an elusively difficult step for the reader to supply. In this paper we prove the Plancherel theorem, essentially, by proving a special case of Theorem 11. We close by deriving a version of Theorem 11 from the Plancherel theorem.
On polarized surfaces (𝑋,𝐿) with ℎ⁰(𝐿)>0, 𝜅(𝑋)=2, and 𝑔(𝐿)=𝑞(𝑋)
Authors: Fukuma Yoshiaki
Published: July 2002

Let

X
X

be a smooth projective surface over

C

\mathbb {C}

and

L
L

an ample Cartier divisor on

X
X

. If the Kodaira dimension

κ
(
X
)
≤
1

\kappa (X)\leq 1

or

dim
⁡

H

0

(
L
)
>
0

\operatorname {dim}H^{0}(L)>0

, the author proved

g
(
L
)
≥
q
(
X
)

g(L)\geq q(X)

, where

q
(
X
)
=
dim
⁡

H

1

(

O

X

)

q(X)=\operatorname {dim}H^{1}(\mathcal {O}_{X})

. If

κ
(
X
)
≤
1

\kappa (X)\leq 1

, then the author studied

(
X
,
L
)

(X,L)

with

g
(
L
)
=
q
(
X
)

g(L)=q(X)

. In this paper, we study the polarized surface

(
X
,
L
)

(X,L)

with

κ
(
X
)
=
2

\kappa (X)=2

,

g
(
L
)
=
q
(
X
)

g(L)=q(X)

, and

dim
⁡

H

0

(
L
)
>
0

\operatorname {dim}H^{0}(L)>0

.
Essential laminations in 𝐼-bundles
Authors: Brittenham Mark
Published: July 2002

We show that, with a few familiar exceptions, every essential lamination in an interval-bundle over a closed surface can be isotoped to lie everywhere transverse to the

I
I

-fibers of the bundle.
On the strong equality between supercompactness and strong compactness
Authors: Apter Arthur , Shelah Saharon
Published: July 2002

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if

V
⊨

V \models

ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension

V
[

G

]
⊨

V[{G}] \models

ZFC + GCH in which, (a) (preservation) for

κ
≤
λ

\kappa \le \lambda

regular, if

V
⊨

V \models {}

“

κ
\kappa

is

λ
\lambda

supercompact”, then

V
[
G
]
⊨

V[G] \models {}

“

κ
\kappa

is

λ
\lambda

supercompact” and so that, (b) (equivalence) for

κ
≤
λ

\kappa \le \lambda

regular,

V
[

G

]
⊨

V[{G}] \models {}

“

κ
\kappa

is

λ
\lambda

strongly compact” iff

V
[

G

]
⊨

V[{G}] \models {}

“

κ
\kappa

is

λ
\lambda

supercompact”, except possibly if

κ
\kappa

is a measurable limit of cardinals which are

λ
\lambda

supercompact.
On composite twisted unknots
Authors: Goodman-Strauss Chaim
Published: July 2002

Following Mathieu, Motegi and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular

V
V

-twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular

V
V

-twisting, or one summand was unaffected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given.
Only single twists on unknots can produce composite knots
Authors: Hayashi Chuichiro , Motegi Kimihiko
Published: July 2002

Let

K
K

be a knot in the

3
3

-sphere

S

3

S^{3}

, and

D
D

a disc in

S

3

S^{3}

meeting

K
K

transversely more than once in the interior. For non-triviality we assume that

|
K
∩
D
|
≥
2

\vert K \cap D \vert \ge 2

over all isotopy of

K
K

. Let

K

n

K_{n}

(

⊂

S

3

\subset S^{3}

) be a knot obtained from

K
K

by cutting and

n
n

-twisting along the disc

D
D

(or equivalently, performing

1

/

n

1/n

-Dehn surgery on

∂
D

\partial D

). Then we prove the following: (1) If

K
K

is a trivial knot and

K

n

K_{n}

is a composite knot, then

|
n
|
≤
1

\vert n \vert \le 1

; (2) if

K
K

is a composite knot without locally knotted arc in

S

3

−
∂
D

S^{3} - \partial D

and

K

n

K_{n}

is also a composite knot, then

|
n
|
≤
2

\vert n \vert \le 2

. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.
Semidirect products of regular semigroups
Authors: Jones Peter , Trotter Peter
Published: July 2002

Within the usual semidirect product

S
∗
T

S*T

of regular semigroups

S
S

and

T
T

lies the set

Reg

(
S
∗
T
)

\text {Reg}\, (S*T)

of its regular elements. Whenever

S
S

or

T
T

is completely simple,

Reg

(
S
∗
T
)

\text {Reg}\, (S*T)

is a (regular) subsemigroup. It is this ‘product’ that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes,

U

{\mathbf U}

and

V

{\mathbf V}

, the e-variety

U

∗

V

{\mathbf U}*{\mathbf V}

generated by

{
Reg

(
S
∗
T
)
:
S
∈

U

,
T
∈

V

}

\{\text {Reg}\, (S*T) : S \in {\mathbf U} , T \in {\mathbf V} \}

is well defined if and only if either

U

{\mathbf U}

or

V

{\mathbf V}

is contained within the e-variety

C
S

{\mathbf {CS}}

of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety

L

I

L{\mathbf I}

of locally inverse semigroups is decomposed as

I

∗

R
Z

{\mathbf I} * {\mathbf {RZ}}

, where

I

{\mathbf I}

is the variety of inverse semigroups and

R
Z

{\mathbf {RZ}}

is that of right zero semigroups; and the e-variety

E
S

{\mathbf {ES}}

of

E
E

-solid semigroups is decomposed as

C
R

∗

G

{\mathbf {CR}}*{\mathbf G}

, where

C
R

{\mathbf {CR}}

is the variety of completely regular semigroups and

G

{\mathbf G}

is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products

U

∗

V

{\mathbf U} * {\mathbf V}

of the above type, as a semidirect product of e-free semigroups from

U

{\mathbf U}

and

V

{\mathbf V}

, “cut down to regular generators”. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups,

E
E

-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups,

E
E

-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different ‘semidirect’ product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.
Virtually free groups with finitely many outer automorphisms
Authors: Pettet Martin
Published: July 2002

Let

G
G

be a finitely generated virtually free group. From a presentation of

G
G

as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for the outer automorphism group of

G
G

to be finite. Two versions of the characterization are given, both effectively verifiable from the graph of groups. The more purely group theoretical criterion is expressed in terms of the structure of the normalizers of the edge groups (Theorem 5.10); the other version involves certain finiteness conditions on the associated

G
G

-tree (Theorem 5.16). Coupled with an earlier result, this completes a description of the finitely generated groups whose full automorphism groups are virtually free.
Twin trees and 𝜆_{Λ}-gons
Authors: Bennett Curtis
Published: July 2002

We define a natural generalization of generalized

n
n

-gons to the case of

Λ
\Lambda

-graphs (where

Λ
\Lambda

is a totally ordered abelian group and

0
>
λ
∈
Λ

0>\lambda \in \Lambda

). We term these objects

λ

Λ

\lambda _{\Lambda }

-gons. We then show that twin trees as defined by Ronan and Tits can be viewed as

(
1
,
0

)

Λ

(1,0)_{\Lambda }

-gons, where

Λ
=
Z
×
Z

\Lambda = Z \times Z

is ordered lexicographically. This allows us to then generalize twin trees to the case of

Λ
\Lambda

-trees. Finally, we give a free construction of

λ

Λ

\lambda _{\Lambda }

-gons in the cases where

Λ
\Lambda

is discrete and has a subgroup of index 

2
2

that does not contain the minimal element of

Λ
\Lambda

.
The homotopy groups of the 𝐿₂-localized Toda-Smith spectrum 𝑉(1) at the prime 3
Authors: Shimomura Katsumi
Published: July 2002

In this paper, we try to compute the homotopy groups of the

L
2

L_2

-localized Toda-Smith spectrum

V
(
1
)

V(1)

at the prime 3 by using the Adams-Novikov spectral sequence, and have almost done so. This computation involves non-trivial differentials

d
5

d_5

and

d
9

d_9

of the Adams-Novikov spectral sequence, different from the case

p
>
3

p>3

. We also determine the homotopy groups of some

L
2

L_2

-localized finite spectra relating to

V
(
1
)

V(1)

. We further show some of the non-trivial differentials on elements relating so-called

β
\beta

-elements in the Adams-Novikov spectral sequence for

π
∗

(

S
0

)

\pi _*(S^0)

.
Dehn surgery on knots in solid tori creating essential annuli
Authors: Hayashi Chuichiro , Motegi Kimihiko
Published: July 2002

Let

M
M

be a

3
3

-manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when

M
M

contains a separating essential annulus. It is shown that

M
M

does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the

3
3

-sphere which are not periodic knots of period

2
2

satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the

2
2

-sphere with exactly three exceptional fibres.
Primitive higher order embeddings of abelian surfaces
Authors: Bauer Th. , Szemberg T.
Published: July 2002

In recent years several concepts of higher order embeddings have been studied:

k
k

-spannedness,

k
k

-very ampleness and

k
k

-jet ampleness. In the present note we consider primitive line bundles on abelian surfaces and give numerical criteria which allow to check whether a given ample line bundle satisfies these properties.
Examples of asymptotic ℓ₁ Banach spaces
Authors: Argyros S. , Deliyanni I.
Published: July 2002

Two examples of asymptotic

ℓ

1

\ell _{1}

Banach spaces are given. The first,

X

u

X_{u}

, has an unconditional basis and is arbitrarily distortable. The second,

X
X

, does not contain any unconditional basic sequence. Both are spaces of the type of Tsirelson’s.
On Poincaré Type Inequalities
Authors: Chen Roger , Li Peter
Published: July 2002

Using estimates of the heat kernel we prove a Poincaré inequality for star-shape domains on a complete manifold. The method also gives a lower bound for the gap of the first two Neumann eigenvalues of a Schrödinger operator.
Degenerations of K3 surfaces in projective space
Authors: Gallego Francisco , Purnaprajna B.
Published: July 2002

The purpose of this article is to study a certain kind of numerical K3 surfaces, the so-called K3 carpets. These are double structures on rational normal scrolls with trivial dualizing sheaf and irregularity

0
0

. As is deduced from our study, K3 carpets can be obtained as degenerations of smooth K3 surfaces. We also study the Hilbert scheme near the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. Contrary to the case of canonical ribbons, not all K3 carpets are smooth points of the Hilbert scheme.
The Jacobson radical of group rings of locally finite groups
Authors: Passman D.
Published: July 2002

This paper is the final installment in a series of articles, started in 1974, which study the semiprimitivity problem for group algebras

K
[
G
]

K[G]

of locally finite groups. Here we achieve our goal of describing the Jacobson radical

J

K
[

G

]

{\mathcal {J}}K[{G}]

in terms of the radicals

J

K
[

A

]

{\mathcal {J}}K[{A}]

of the group algebras of the locally subnormal subgroups

A
A

of

G
G

. More precisely, we show that if

char
⁡
K
=
p
>
0

\operatorname {char} K=p>0

and if

O

p

(
G
)
=
1

\mathbb {O}_{p}(G)=1

, then the controller of

J

K
[

G

]

{\mathcal {J}}K[{G}]

is the characteristic subgroup

S

p

(
G
)

\mathbb {S}^{p}(G)

generated by the locally subnormal subgroups

A
A

of

G
G

with

A
=

O

p
′

(
A
)

A=\mathbb {O}^{p’}(A)

. In particular, we verify a conjecture proposed some twenty years ago and, in so doing, we essentially solve one half of the group ring semiprimitivity problem for arbitrary groups. The remaining half is the more difficult case of finitely generated groups. This article is effectively divided into two parts. The first part, namely the material in Sections 2–6, covers the group theoretic aspects of the proof and may be of independent interest. The second part, namely the work in Sections 7–12, contains the group ring and ring theoretic arguments and proves the main result. As usual, it is necessary for us to work in the more general context of twisted group algebras and crossed products. Furthermore, the proof ultimately depends upon results which use the Classification of the Finite Simple Groups.
Tangential flatness and global rigidity of higher rank lattice actions
Authors: Qian Nantian
Published: July 2002

We establish the continuous tangential flatness for orientable weakly Cartan actions of higher rank lattices. As a corollary, we obtain the global rigidity of Anosov Cartan actions.
Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces
Authors: Biswas Indranil , Arés-Gastesi Pablo , Govindarajan Suresh
Published: July 2002

In this paper we study the relation between parabolic Higgs vector bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL(

k
,

R

k,\mathbb R

). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of

k
=
2

k=2

is the usual Teichmüller space of the punctured surface.
On the rational homotopy type of function spaces
Authors: Brown Edgar , Szczarba Robert
Published: July 2002

The main result of this paper is the construction of a minimal model for the function space

F

(
X
,
Y
)

\mathcal {F}(X,Y)

of continuous functions from a finite type, finite dimensional space

X
X

to a finite type, nilpotent space

Y
Y

in terms of minimal models for

X
X

and

Y
Y

. For the component containing the constant map,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript asterisk Baseline left-parenthesis script upper F left-parenthesis upper X comma upper Y right-parenthesis right-parenthesis circled-times upper Q equals pi Subscript asterisk Baseline left-parenthesis upper Y right-parenthesis circled-times upper H Superscript minus asterisk Baseline left-parenthesis upper X semicolon upper Q right-parenthesis">

π

∗

(

F

(
X
,
Y
)
)
⊗
Q
=

π

∗

(
Y
)
⊗

H

−
∗

(
X
;
Q
)

\pi _{*}(\mathcal {F}(X,Y))\otimes Q =\pi _{*}(Y)\otimes H^{-*}(X;Q)

in positive dimensions. When

X
X

is formal, there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for

Y
Y

and the coproduct of

H

∗

(
X
;
Q
)

H_{*}(X;Q)

. We also give a version of the main result for the space of cross sections of a fibration.
Lower bounds for derivatives of polynomials and Remez type inequalities
Authors: Erdélyi Tamás , Nevai Paul
Published: July 2002

P. Turán [Über die Ableitung von Polynomen, Comositio Math. 7 (1939), 89–95] proved that if all the zeros of a polynomial

p
p

lie in the unit interval

I

=
def

[
−
1
,
1
]

I \overset {\text {def}}{=} [-1,1]

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar p prime double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper I right-parenthesis Baseline greater-than-or-equal-to StartRoot degree left-parenthesis p right-parenthesis EndRoot slash 6 double-vertical-bar p double-vertical-bar Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis upper I right-parenthesis">

‖

p
′

‖

L

∞

(
I
)

≥

deg
⁡
(
p
)

/

6

‖
p

‖

L

∞

(
I
)

\|p’\|_{L^{\infty }(I)}\ge {\sqrt {\deg (p)}}/{6}\; \|p\|_{L^{\infty }(I)}\;

. Our goal is to study the feasibility of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript n right-arrow normal infinity Endscripts double-vertical-bar p prime Subscript n double-vertical-bar Subscript upper X Baseline slash double-vertical-bar p Subscript n Baseline double-vertical-bar Subscript upper Y Baseline equals normal infinity">

lim

n
→
∞

‖

p

n

′

‖

X

/

‖

p

n

‖

Y

=
∞

\lim _{{n\to \infty } }{\|p_{n}’\|_{X}} / {\|p_{n}\|_{Y}} =\infty

for sequences of polynomials

{

p

n

}

n
∈

N

\{p_{n}\}_{n\in \mathbb N }

whose zeros satisfy certain conditions, and to obtain lower bounds for derivatives of (generalized) polynomials and Remez type inequalities for generalized polynomials in various spaces.
Mean-boundedness and Littlewood-Paley for separation-preserving operators
Authors: Berkson Earl , Gillespie T.
Published: July 2002

Suppose that

(
Ω
,

M

,
μ
)

(\Omega ,\mathcal {M},\mu )

is a

σ
\sigma

-finite measure space,

1
>
p
>
∞

1>p>\infty

, and

T
:

L

p

(
μ
)
→

L

p

(
μ
)

T: L^{p}(\mu )\to L^{p}(\mu )

is a bounded, invertible, separation-preserving linear operator such that the linear modulus of

T
T

is mean-bounded. We show that

T
T

has a spectral representation formally resembling that for a unitary operator, but involving a family of projections in

L

p

(
μ
)

L^{p}(\mu )

which has weaker properties than those associated with a countably additive Borel spectral measure. This spectral decomposition for

T
T

is shown to produce a strongly countably spectral measure on the “dyadic sigma-algebra” of

T

\mathbb {T}

, and to furnish

L

p

(
μ
)

L^{p}(\mu )

with abstract analogues of the classical Littlewood-Paley and Vector-Valued M. Riesz Theorems for

ℓ

p

(

Z

)

\ell ^{p}(\mathbb {Z})

.
Proximity inequalities and bounds for the degree of invariant curves by foliations of ℙ_{ℂ}²
Authors: Campillo Antonio , Carnicer Manuel
Published: July 2002

In this paper we prove that if

C
C

is a reduced curve which is invariant by a foliation

F

\mathcal F

in the complex projective plane then one has

∂

∘
_

C
≤

∂

∘
_

F

+
2
+
a

\partial ^{\underline {\circ }} C\leq \partial ^{\underline {\circ }} \mathcal F+2+a

where

a
a

is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If

F

\mathcal F

is nondicritical or if

C
C

has only nodes as singularities, then one gets

a
=
0

a=0

and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.
Signed Quasi-Measures
Authors: Grubb D.
Published: July 2002

Let

X
X

be a compact Hausdorff space and let

A

\mathcal {A}

denote the subsets of

X
X

which are either open or closed. A quasi-linear functional is a map

ρ
:
C
(
X
)
→

R

\rho :C(X)\rightarrow \mathbb {R}

which is linear on singly generated subalgebras and such that

|

ρ
(
f
)

|

≤
M
‖
f
‖

|\rho (f)|\leq M\|f\|

for some

M
>
∞

M>\infty

. There is a one-to-one correspondence between the quasi-linear functional on

C
(
X
)

C(X)

and the set functions

μ
:

A

→

R

\mu :\mathcal {A}\rightarrow \mathbb {R}

such that i)

μ
(
∅
)
=
0

\mu (\emptyset )=0

, ii) If

A
,
B
,
A
∪
B
∈

A

A,B,A\cup B\in \mathcal {A}

with

A
A

and

B
B

disjoint, then

μ
(
A
∪
B
)
=
μ
(
A
)
+
μ
(
B
)

\mu (A\cup B)=\mu (A)+\mu (B)

, iii) There is an

M
>
∞

M>\infty

such that whenever

{

U
α

}

\{U_\alpha \}

are disjoint open sets,

∑

|

μ
(

U
α

)

|

≤
M

\sum |\mu (U_\alpha )|\leq M

, and iv) if

U
U

is open and

ε
>
0

\varepsilon >0

, there is a compact

K
⊆
U

K\subseteq U

such that whenever

V
⊆
U
∖
K

V\subseteq U\setminus K

is open, then

|

μ
(
V
)

|

>
ε

|\mu (V)|>\varepsilon

. The space of quasi-linear functionals is investigated and quasi-linear maps between two

C
(
X
)

C(X)

spaces are studied.
Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties
Authors: Caenepeel S. , Militaru G. , Zhu Shenglin
Published: July 2002

We study the following question: when is the right adjoint of the forgetful functor from the category of

(
H
,
A
,
C
)

(H,A,C)

-Doi-Hopf modules to the category of

A
A

-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that

C
⊗
A

C\otimes A

and the smash product

A
#

C
∗

A\# C^*

are isomorphic as

(
A
,
A
#

C
∗

)

(A, A\# C^*)

-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case

A
=
H

A=H

, and this leads to the notion of

k
k

-Frobenius

H
H

-module coalgebra. In the special case of Yetter-Drinfel′d modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if

H
H

is finite dimensional and unimodular.
Decomposition theorems and approximation by a “floating" system of exponentials
Authors: Belinskii E.
Published: July 2002

The main problem considered in this paper is the approximation of a trigonometric polynomial by a trigonometric polynomial with a prescribed number of harmonics. The method proposed here gives an opportunity to consider approximation in different spaces, among them the space of continuous functions, the space of functions with uniformly convergent Fourier series, and the space of continuous analytic functions. Applications are given to approximation of the Sobolev classes by trigonometric polynomials with prescribed number of harmonics, and to the widths of the Sobolev classes. This work supplements investigations by Maiorov, Makovoz and the author where similar results were given in the integral metric.
Algebraic transition matrices in the Conley index theory
Authors: Franzosa Robert , Mischaikow Konstantin
Published: July 2002

We introduce the concept of an algebraic transition matrix. These are degree zero isomorphisms which are upper triangular with respect to a partial order. It is shown that all connection matrices of a Morse decomposition for which the partial order is a series-parallel admissible order are related via a conjugation with one of these transition matrices. This result is then restated in the form of an existence theorem for global bifurcations. Simple examples of how these results can be applied are also presented.
Sobolev estimates for operators given by averages over cones
Authors: Cuccagna Scipio
Published: July 2002

We prove a result related to work by A. Greenleaf and G. Uhlmann concerning Sobolev estimates for operators given by averages over cones. This is done using the almost orthogonality lemma of Cotlar and Stein, and the van der Corput lemma on oscillatory integrals.
Equations in a free 𝐐-group
Authors: Kharlampovich O. , Myasnikov A.
Published: July 2002

An algorithm is constructed that decides if a given finite system of equations over a free

Q

\mathbf {Q}

-group has a solution, and if it does, finds a solution.
Comparing Heegaard splittings -the bounded case
Authors: Rubinstein Hyam , Scharlemann Martin
Published: July 2002

In a recent paper we used Cerf theory to compare strongly irreducible Heegaard splittings of the same closed irreducible orientable 3-manifold. This captures all irreducible splittings of non-Haken 3-manifolds. One application is a solution to the stabilization problem for such splittings: If

p
≤
q

p \leq q

are the genera of two splittings, then there is a common stabilization of genus

5
p
+
8
q
−
9

5p + 8q - 9

. Here we show how to obtain similar results even when the 3-manifold has boundary.
Geometric properties of the double-point divisor
Authors: Ilic Bo
Published: July 2002

The locus of double points obtained by projecting a variety

X

n

⊂

P

N

X^{n} \subset \mathbf {P}^N

to a hypersurface in

P

n
+
1

\mathbf {P}^{n+1}

moves in a linear system which is shown to be ample if and only if

X
X

is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined.
Self-duality operators on odd dimensional manifolds
Authors: Fan Houhong
Published: July 2002

In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally freeS1S^{1}-actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author’s previous papers.
Filling by holomorphic curves in symplectic 4-manifolds
Authors: Ye Rugang
Published: July 2002

We develop a general framework for embedded (immersed)

J
J

-holomorphic curves and a systematic treatment of the theory of filling by holomorphic curves in 4-dimensional symplectic manifolds. In particular, a deformation theory and an intersection theory for

J
J

-holomorphic curves with boundary are developed. Bishop’s local filling theorem is extended to almost complex manifolds. Existence and uniqueness of global fillings are given complete proofs. Then they are extended to the situation with nontrivial

J
J

-holomorphic spheres, culminating in the construction of singular fillings.
Tessellations of solvmanifolds
Authors: Witte Dave
Published: July 2002

Let

A
A

be a closed subgroup of a connected, solvable Lie group 

G
G

, such that the homogeneous space

A
∖
G

A\backslash G

is simply connected. As a special case of a theorem of C. T. C. Wall, it is known that every tessellation

A
∖
G

/

Γ

A\backslash G/\Gamma

of

A
∖
G

A\backslash G

is finitely covered by a compact homogeneous space

G
′

/

Γ
′

G’/\Gamma ’

. We prove that the covering map can be taken to be very well behaved — a “crossed" affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on

A
∖
G

/

Γ

A\backslash G/\Gamma

that has a dense orbit is covered by a natural flow on

G
′

/

Γ
′

G’/\Gamma ’

.
New subfactors from braid group representations
Authors: Erlijman Juliana
Published: July 2002

This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II

1

_{1}

factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.
𝐿^{𝑝} and operator norm estimates for the complex time heat operator on homogeneous trees
Authors: Setti Alberto
Published: July 2002

Let

X

\mathfrak {X}

be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator

H

ζ

{\mathcal {H}}_{\zeta }

induced by the natural Laplace operator on

X

\mathfrak {X}

. We prove comparable upper and lower bounds for the

L

p

L^{p}

norms of its convolution kernel

h

ζ

h_{\zeta }

and derive precise estimates for the

L

p

–

L

r

L^{p}\text {–}L^{r}

operator norms of

H

ζ

{\mathcal {H}}_{\zeta }

for

ζ
\zeta

belonging to the half plane

Re

ζ
≥
0.

\text {Re}\,\zeta \geq 0.

In particular, when

ζ
\zeta

is purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on

X

\mathfrak {X}

.
Hypercyclicity in the scattering theory for linear transport equation
Authors: Emamirad H.
Published: July 2002

We show how the hypercyclicity of the transport semigroup can intervene in the scattering theory to characterize the density property of the Lax and Phillips representation theorem and conversely, how the existence of the wave operators of the scattering theory can be used for recovering the hypercyclicity of the absorbing transport group in some weighted

L

1

L^{1}

spaces.
On the equivariant Morse complex of the free loop space of a surface
Authors: Hingston Nancy
Published: July 2002

We prove two theorems about the equivariant topology of the free loop space of a surface. The first deals with the nondegenerate case and says that the “ordinary” Morse complex can be given an

O
(
2
)

O(2)

-action in such a way that it carries the

O
(
2
)

O(2)

-homotopy type of the free loop space. The second says that, in terms of topology, the iterates of an isolated degenerate closed geodesic “look like” the continuous limit of the iterates of a finite, fixed number of nondegenerate closed geodesics.
Berezin’s quantization on flag manifolds and spherical modules
Authors: Karabegov Alexander
Published: July 2002

We show that the theory of spherical Harish-Chandra modules naturally gives rise to Berezin’s symbol quantization on generalized flag manifolds. It provides constructions of symbol algebras and of their representations for covariant and contravariant symbols, and also for symbols which so far have no explicit definition. For all these symbol algebras we give a general proof of the correspondence principle.
Dyadic equivalence to completely positive entropy
Authors: Fieldsteel Adam , Hasfura-Buenaga J.
Published: July 2002

We show that every free ergodic action of

⨁
1
∞

Z

2

\bigoplus _1^\infty {\mathbb Z}_2

of positive entropy is dyadically equivalent to an action with completely positive entropy.
The behavior of the heat operator on weighted Sobolev spaces
Authors: Hile G. , Mawata C.
Published: July 2002

Denoting by

H

{\mathcal {H}}

the heat operator in

R

n
+
1

R^{n+1}

, we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which

H

{\mathcal {H}}

is an injection, a surjection, or an isomorphism. We also describe the range and kernel of

H

{\mathcal {H}}

in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in

R

n

R^{n}

.
Subvarieties of 𝒮𝒰_{𝒞}(2) and 2𝜃-divisors in the Jacobian
Authors: Oxbury W. , Pauly C. , Previato E.
Published: July 2002

We explore some of the interplay between Brill-Noether subvarieties of the moduli space

S
U

C

(
2
,
K
)

{\mathcal {SU}}_C(2,K)

of rank 2 bundles with canonical determinant on a smooth projective curve and

2
θ

2\theta

-divisors, via the inclusion of the moduli space into

|

2
Θ

|

|2\Theta |

, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus

>
6

>6

, though no higher, bundles with

>
2

>2

sections are cut out by

Γ

00

\Gamma _{00}

; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.
Convergence of random walks on the circle generated by an irrational rotation
Authors: Su Francis
Published: July 2002

Fix

α
∈
[
0
,
1
)

\alpha \in [0,1)

. Consider the random walk on the circle

S
1

S^1

which proceeds by repeatedly rotating points forward or backward, with probability

1
2

\frac 12

, by an angle

2
π
α

2\pi \alpha

. This paper analyzes the rate of convergence of this walk to the uniform distribution under “discrepancy” distance. The rate depends on the continued fraction properties of the number

ξ
=
2
α

\xi =2\alpha

. We obtain bounds for rates when

ξ
\xi

is any irrational, and a sharp rate when

ξ
\xi

is a quadratic irrational. In that case the discrepancy falls as

k

−

1
2

k^{-\frac 12}

(up to constant factors), where

k
k

is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of

ξ
\xi

which allows for tighter bounds on terms which appear in the Erdős-Turán inequality.
Induction theorems on the stable rationality of the center of the ring of generic matrices
Authors: Beneish Esther
Published: July 2002

Following Procesi and Formanek, the center of the division ring of

n
×
n

n\times n

generic matrices over the complex numbers

C

\mathbf C

is stably equivalent to the fixed field under the action of

S
n

S_n

, of the function field of the group algebra of a

Z

S
n

ZS_n

-lattice, denoted by

G
n

G_n

. We study the question of the stable rationality of the center

C
n

C_n

over the complex numbers when

n
n

is a prime, in this module theoretic setting. Let

N
N

be the normalizer of an

n
n

-sylow subgroup of

S
n

S_n

. Let

M
M

be a

Z

S
n

ZS_n

-lattice. We show that under certain conditions on

M
M

, induction-restriction from

N
N

to

S
n

S_n

does not affect the stable type of the corresponding field. In particular,

C

(

G
n

)

\mathbf C (G_n)

and

C

(
Z
G

⊗

Z
N

G
n

)

\mathbf C(ZG\otimes _{ZN}G_n)

are stably isomorphic and the isomorphism preserves the

S
n

S_n

-action. We further reduce the problem to the study of the localization of

G
n

G_n

at the prime

n
n

; all other primes behave well. We also present new simple proofs for the stable rationality of

C
n

C_n

over

C

\mathbf C

, in the cases

n
=
5

n=5

and

n
=
7

n=7

.
Witten-Helffer-Sjöstrand theory for 𝑆¹-equivariant cohomology
Authors: Wai Hon-kit
Published: July 2002

Given an

S
1

S^1

-invariant Morse function

f
f

and an

S
1

S^1

-invariant Riemannian metric

g
g

, a family of finite dimensional subcomplexes
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ModifyingAbove normal upper Omega With tilde Subscript i n v comma s m Superscript asterisk Baseline left-parenthesis upper M comma t right-parenthesis comma upper D left-parenthesis t right-parenthesis right-parenthesis">

(

Ω
~

i
n
v
,
s
m

∗

(
M
,
t
)
,
D
(
t
)
)

(\widetilde \Omega ^*_{inv,sm}(M,t), D(t))

,

t
∈
[
0
,
∞
)

t\in [0,\infty )

, of the Witten deformation of the

S
1

S^1

-equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove normal upper Delta With tilde Superscript k Baseline left-parenthesis t right-parenthesis equals upper D Subscript k Superscript asterisk Baseline left-parenthesis t right-parenthesis upper D Subscript k Baseline left-parenthesis t right-parenthesis plus upper D Subscript k minus 1 Baseline left-parenthesis t right-parenthesis upper D Subscript k minus 1 Superscript asterisk Baseline left-parenthesis t right-parenthesis">

Δ
~

k

(
t
)
=

D
k
∗

(
t
)

D
k

(
t
)
+

D

k
−
1

(
t
)

D

k
−
1

∗

(
t
)

\widetilde \Delta ^k(t)=D^*_k(t)D_k(t)+D_{k-1}(t)D^*_{k-1}(t)

as

t
→
∞

t\to \infty

. In fact the spectrum of

Δ
~

k

(
t
)

\widetilde \Delta ^k(t)

can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis ModifyingAbove normal upper Omega With tilde Subscript i n v comma s m Superscript asterisk Baseline left-parenthesis upper M comma t right-parenthesis comma upper D left-parenthesis t right-parenthesis right-parenthesis">

(

Ω
~

i
n
v
,
s
m

∗

(
M
,
t
)
,
D
(
t
)
)

( \widetilde \Omega ^*_{inv,sm}(M,t),D(t))

as the complex of eigenforms corresponding to the small eigenvalues of

Δ
~

(
t
)

\widetilde \Delta (t)

. This permits us to verify the

S
1

S^1

-equivariant Morse inequalities. Moreover suppose

f
f

is self-indexing and

(
f
,
g
)

(f,g)

satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as

t
→
∞

t\to \infty

to a geometric complex which is induced by

(
f
,
g
)

(f,g)

and calculates the

S
1

S^1

-equivariant cohomology of

M
M

.
Properties of some families of hypergeometric orthogonal polynomials in several variables
Authors: van Diejen J.
Published: July 2002

Limiting cases are studied of the Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials. We recover recently and not so recently introduced families of hypergeometric orthogonal polynomials in several variables consisting of multivariable Wilson, continuous Hahn and Jacobi type polynomials, respectively. For each class of polynomials we provide systems of difference (or differential) equations, recurrence relations, and expressions for the (squared) norms of the polynomials in question.
Equations in the Q-completion of a torsion-free hyperbolic group
Authors: Kharlampovich O. , Lioutikova E. , Myasnikov A.
Published: July 2002

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group.
Vertex operators for twisted quantum affine algebras
Authors: Jing Naihuan , Misra Kailash
Published: July 2002

We construct explicitly the

q
q

-vertex operators (intertwining operators) for the level one modules

V
(

Λ
i

)

V(\Lambda _i)

of the classical quantum affine algebras of twisted types using interacting bosons, where

i
=
0
,
1

i=0, 1

for

A

2
n
−
1

(
2
)

A_{2n-1}^{(2)}

(

n
≥
3

n\geq 3

),

i
=
0

i=0

for

D
4

(
3
)

D_4^{(3)}

,

i
=
0
,
n

i=0, n

for

D

n
+
1

(
2
)

D_{n+1}^{(2)}

(

n
≥
2

n\geq 2

), and

i
=
n

i=n

for

A

2
n

(
2
)

A_{2n}^{(2)}

(

n
≥
1

n\geq 1

). A perfect crystal graph for

D
4

(
3
)

D_4^{(3)}

is constructed as a by-product.
Harnack inequality for the linearized parabolic Monge-Ampère equation
Authors: Huang Qingbo
Published: July 2002

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation
\[

u

t

−
tr
(
(

D

2

ϕ
(
x
)

)

−
1

D

2

u
)
=
0

u_{t}-\text {tr}((D^{2}\phi (x))^{-1}D^{2}u)=0

\]
on parabolic sections associated with

ϕ
(
x
)

\phi (x)

, under the assumption that the Monge-Ampère measure generated by

ϕ
\phi

satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group

A
T
(
n
)
×
A
T
(
1
)

AT(n)\times AT(1)

, where

A
T
(
n
)

AT(n)

denotes the group of all invertible affine transformations on

R

n

{\mathbf {R}}^{n}

.