bluuebunny/crossref_metadata_2025 huggingface dataset:
Dual decompositions of 4-manifolds
Authors: Quinn Frank
Published: July 2002

This paper concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index≤2\leq 2. In dimensions≥5\geq 5results of Smale (trivialπ1\pi _{1}) and Wall (generalπ1\pi _{1}) describe analogous decompositions up to diffeomorphism in terms of homotopy type of skeleta or chain complexes. In dimension 4 we show the same data determines decompositions up to 2-deformation of their spines. In higher dimensions spine 2-deformation implies diffeomorphism, but in dimension 4 the fundamental group of the boundary is not determined. Sample results: (1.5) Two 2-complexes are (up to 2-deformation) spines of a dual decomposition of the 4-sphere if and only if they satisfy the conclusions of Alexander-Lefshetz duality (H1K≃H2LH_{1}K\simeq H^{2}LandH2K≃H1LH_{2}K\simeq H^{1}L). (3.3) If(N,∂N)(N,\partial N)is 1-connected then there is a “pseudo” handle decomposition without 1-handles, in the sense that there is a pseudo collar(M,∂N)(M,\partial N)(a relative 2-handlebody with spine that 2-deforms to∂N\partial N) andNNis obtained from this by attaching handles of index≥2\geq 2.
On crepant resolution of some hypersurface singularities and a criterion for UFD
Authors: Lin Hui-Wen
Published: July 2002

In this article, we find some diagonal hypersurfaces that admit crepant resolutions. We also give a criterion for unique factorization domains.
Trudinger inequalities without derivatives
Authors: MacManus Paul , Pérez Carlos
Published: July 2002

We prove that the Trudinger inequality holds on connected homogeneous spaces for functions satisfying a very weak type of Poincaré inequality. We also illustrate the connection between this result and the John-Nirenberg theorem for BMO.
Invertibility preserving linear maps on ℒ(𝒳)
Authors: Sourour A.
Published: July 2002

For Banach spaces

X
X

and

Y
Y

, we show that every unital bijective invertibility preserving linear map between

L

(
X
)

\mathcal L(X)

and

L

(
Y
)

\mathcal L(Y)

is a Jordan isomorphism. The same conclusion holds for maps between

C

I
+

K

(
X
)

\mathbb CI+ \mathcal K(X)

and

C

I
+

K

(
Y
)

\mathbb CI+\mathcal K(Y)

.
Smooth classification of geometrically finite one-dimensional maps
Authors: Jiang Yunping
Published: July 2002

The scaling function of a one-dimensional Markov map is defined and studied. We prove that the scaling function of a non-critical geometrically finite one-dimensional map is Hölder continuous, while the scaling function of a critical geometrically finite one-dimensional map is discontinuous. We prove that scaling functions determine Lipschitz conjugacy classes, and moreover, that the scaling function and the exponents and asymmetries of a geometrically finite one-dimensional map are complete

C

1

C^{1}

-invariants within a mixing topological conjugacy class.
On vanishing of characteristic numbers in Poincaré complexes
Authors: Byun Yanghyun
Published: July 2002

Let

G
r

(
X
)
⊂

π
r

(
X
)

G_r(X)\subset \pi _r(X)

be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map

G
r

(
X
)
→

H
r

(
X
;
R
)

G_r(X)\rightarrow H_r(X; R)

is non-trivial and

R
R

is a field. We will prove: if

X
X

is a Poincaré complex oriented in

R
R

-coefficient, all the characteristic numbers of

X
X

in

R
R

-coefficient vanish. Similarly, if

R
=

Z
p

R=Z_p

and

X
X

is a

Z
p

Z_p

-Poincaré complex, then all the mod

p
p

Wu numbers vanish. We will also show that the existence of a non-trivial derivation on

H
∗

(
X
;

Z
p

)

H^*(X; Z_p)

with some suitable conditions implies vanishing of mod

p
p

Wu numbers.
Complex convexity in Lebesgue-Bochner Function Spaces
Authors: Dowling Patrick , Hu Zhibao , Mupasiri Douglas
Published: July 2002

Complex geometric properties of continuously quasi-normed spaces are introduced and their relationship to their analogues in real Banach spaces is discussed. It is shown that these properties lift from a continuously quasi-normed space

X
X

to

L
p

(
μ
,
X
)

L^p(\mu , X)

, for

0
>
p
>
∞

0 > p > \infty

. Local versions of these properties and results are also considered.
Hyperbolic groups and their quotients of bounded exponents
Authors: Ivanov S. , Ol’shanskii A.
Published: July 2002

In 1987, Gromov conjectured that for every non-elementary hyperbolic group

G
G

there is an

n
=
n
(
G
)

n =n(G)

such that the quotient group

G

/

G

n

G/G^{n}

is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of

G

/

G

n

G/G^{n}

is given, it is proven that the word and conjugacy problem are solvable in

G

/

G

n

G/G^{n}

and that

⋂

k
=
1

∞

G

k

=
{
1
}

\bigcap _{k=1}^{\infty }G^{k} = \{ 1\}

. The proofs heavily depend upon prior authors’ results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.
Weakly repelling fixpoints and the connectivity of wandering domains
Authors: Bergweiler Walter , Terglane Norbert
Published: July 2002

It is proved that if a transcendental meromorphic functionffhas a multiply-connected wandering domain, thenffhas a fixpointz0z_0such that|f′(z0)|>1|f’(z_0)|>1orf′(z0)=1f’(z_0)=1. Entire functions with a multiply-connected wandering domain have infinitely many such fixpoints. These results are used to show that solutions of certain differential equations do not have wandering domains at all.
Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions
Authors: Aikawa H. , Borichev A.
Published: July 2002

We show that if a set

E
E

is dispersely decomposed into subsets, then the capacity of

E
E

is comparable to the summation of the capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of

E
E

. These properties hold for classical capacities,

L

p

L^{p}

-capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points.
𝑆-integral points of ℙⁿ-{2𝕟+1 hyperplanes in general position over number fields and function fields}
Authors: Wang Julie
Published: July 2002

For the number field case we will give an upper bound on the number of the

S
S

-integral points in

P

n

(
K
)
−
{
2
n
+
1
 hyperplanes in general

\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general}

position
}

\text {position}\}

. The main tool here is the explicit upper bound of the number of solutions of

S
S

-unit equations (Invent. Math. 102 (1990), 95–107). For the function field case we will give a bound on the height of the

S
S

-integral points in

P

n

(
K
)
−
{
2
n
+
1
 hyperplanes in general position
}

\mathbb {P}^n(K)-\{ 2n+1\text { hyperplanes in general position}\}

. We will also give a bound for the number of “generators" of those

S
S

-integral points. The main tool here is the

S
S

-unit Theorem by Brownawell and Masser (Proc. Cambridge Philos. Soc. 100 (1986), 427–434).
Global (and local) analyticity for second order operators constructed from rigid vector fields on products of tori
Authors: Tartakoff David
Published: July 2002

We prove global analytic hypoellipticity on a product of tori for partial differential operators which are constructed as rigid (variable coefficient) quadratic polynomials in real vector fields satisfying the Hörmander condition and where

P
P

satisfies a “maximal” estimate. We also prove an analyticity result that is local in some variables and global in others for operators whose prototype is
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P equals left-parenthesis StartFraction partial-differential Over partial-differential x 1 EndFraction right-parenthesis squared plus left-parenthesis StartFraction partial-differential Over partial-differential x 2 EndFraction right-parenthesis squared plus left-parenthesis a left-parenthesis x 1 comma x 2 right-parenthesis StartFraction partial-differential Over partial-differential t EndFraction right-parenthesis squared">

P
=

(

∂

∂

x
1

)

2

+

(

∂

∂

x
2

)

2

+

(
a
(

x
1

,

x
2

)

∂

∂
t

)

2

P=\left ( \frac \partial {\partial x_1}\right ) ^2+\left ( \frac \partial { \partial x_2}\right ) ^2+\left ( a(x_1,x_2)\frac \partial {\partial t}\right )^2

\]
(with analytic

a
(
x
)
,
a
(
0
)
=
0

a(x),a(0)=0

, naturally, but not identically zero). The results, because of the flexibility of the methods, generalize recent work of Cordaro and Himonas in [Global analytic hypoellipticity of a class of degenerate elliptic operators on the torus, Math. Res. Lett. 1 (1994), 501–510] and Himonas in [On degenerate elliptic operators of infinite type, Math. Z. (to appear)] which showed that certain operators known not to be locally analytic hypoelliptic (those of Baouendi and Goulaouic [Analyticity for degenerate elliptic equations and applications, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1971, pp. 79–84], Hanges and Himonas [Singular solutions for sums of squares of vector fields, Comm. Partial Differential Equations 16 (1991), 1503–1511], and Christ [Certain sums of squares of vector fields fail to be analytic hypoelliptic, Comm. Partial Differential Equations 10 (1991), 1695–1707]) were globally analytic hypoelliptic on products of tori.
Simplifying stable mappings into the plane from a global viewpoint
Authors: Kobayashi Mahito , Saeki Osamu
Published: July 2002

Let

f
:
M
→

R

2

f : M \to \mathbf {R}^{2}

be a

C

∞

C^{\infty }

stable map of an

n
n

-dimensional manifold into the plane. The main purpose of this paper is to define a global surgery operation on

f
f

which simplifies the configuration of the critical value set and which does not change the diffeomorphism type of the source manifold

M
M

. For this purpose, we also study the quotient space

W

f

W_{f}

of

f
f

, which is the space of the connected components of the fibers of

f
f

, and we completely determine its local structure for arbitrary dimension

n
n

of the source manifold

M
M

. This is a completion of the result of Kushner, Levine and Porto for dimension 3 and that of Furuya for orientable manifolds of dimension 4. We also pay special attention to dimension 4 and obtain a simplification theorem for stable maps whose regular fiber is a torus or a 2-sphere, which is a refinement of a result of Kobayashi.
Spatial chaotic structure of attractors of reaction-diffusion systems
Authors: Afraimovich V. , Babin A. , Chow S.-N.
Published: July 2002

The dynamics described by a system of reaction-diffusion equations with a nonlinear potential exhibits complicated spatial patterns. These patterns emerge from preservation of homotopy classes of solutions with bounded energies. Chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree. This group corresponds to level sets of the potential. The estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem.
The fixed-point property for simply connected plane continua
Authors: Hagopian Charles
Published: July 2002

We answer a question of R. Mańka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let

M
M

be a plane continuum with the property that every simple closed curve in

M
M

bounds a disk in

M
M

. Then every map of

M
M

that sends each arc component into itself has a fixed point. Hence every deformation of

M
M

has a fixed point. These results are corollaries to the following general theorem. If

M
M

is a plane continuum,

D

\mathcal {D}

is a decomposition of

M
M

, and each element of

D

\mathcal {D}

is simply connected, then every map of

M
M

that sends each element of

D

\mathcal {D}

into itself has a fixed point.
Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature
Authors: Mercuri Francesco , Noronha Maria
Published: July 2002

In this paper we study compact submanifolds of Euclidean space with nonnegative isotropic curvature and low codimension. We determine their homology completely in the case of hypersurfaces and for some low codimensional conformally flat immersions.
Algebraic surfaces with log canonical singularities and the fundamental groups of their smooth parts
Authors: Zhang D.-Q.
Published: July 2002

Let(S,Δ)(S, \Delta )be a log surface with at worst log canonical singularities and reduced boundaryΔ\Deltasuch that−(KS+Δ)-(K_{S}+\Delta )is nef and big. We shall prove thatSo=S−SingS−ΔS^{o} = S - Sing S - \Deltaeither has finite fundamental group or is affine-ruled. Moreover,π1(So)\pi _{1}(S^{o})and the structure ofSSare determined in some sense whenΔ=0\Delta = 0.
Seifert manifolds with fiber spherical space forms
Authors: Lee Jong Bum , Lee Kyung , Raymond Frank
Published: July 2002

We study the Seifert fiber spaces modeled on the product space

S
3

×

R

2

S^3 \times \mathbb {R}^2

. Such spaces are “fiber bundles” with singularities. The regular fibers are spherical space-forms of

S
3

S^3

, while singular fibers are finite quotients of regular fibers. For each of possible space-form groups

Δ
\Delta

of

S
3

S^3

, we obtain a criterion for a group extension

Π
\varPi

of

Δ
\Delta

to act on

S
3

×

R

2

S^3 \times \mathbb {R}^2

as weakly

S
3

S^3

-equivariant maps, which gives rise to a Seifert fiber space modeled on

S
3

×

R

2

S^3 \times \mathbb {R}^2

with weakly

S
3

S^3

-equivariant maps

T
O
P

S
3

(

S
3

×

R

2

)

\mathrm {TOP}_{S^3}(S^3 \times \mathbb {R}^2)

as the universal group. In the course of proving our main results, we also obtain an explicit formula for

H
2

(
Q
;

Z

)

H^2(Q; \mathbb {Z})

for a cocompact crystallographic or Fuchsian group

Q
Q

. Most of our methods for

S
3

S^3

apply to compact Lie groups with discrete center, and we state some of our results in this general context.
An Extension of Lomonosov’s Techniques to Non-compact Operators
Authors: Simonic Aleksander
Published: July 2002

The aim of this work is to generalize Lomonosov’s techniques in order to apply them to a wider class of not necessarily compact operators. We start by establishing a connection between the existence of invariant subspaces and density of what we define as the associated Lomonosov space in a certain function space. On a Hilbert space, approximation with Lomonosov functions results in an extended version of Burnside’s Theorem. An application of this theorem to the algebra generated by an essentially self-adjoint operator

A
A

yields the existence of vector states on the space of all polynomials restricted to the essential spectrum of

A
A

. Finally, the invariant subspace problem for compact perturbations of self-adjoint operators acting on a real Hilbert space is translated into an extreme problem and the solution is obtained upon differentiating certain real-valued functions at their extreme.
On quadratic forms of height two and a theorem of Wadsworth
Authors: Hoffmann Detlev
Published: July 2002

Letφ\varphiandψ\psibe anisotropic quadratic forms over a fieldFFof characteristic not22. Their function fieldsF(φ)F( \varphi )andF(ψ)F(\psi )are said to be equivalent (overFF) ifφ⊗F(ψ)\varphi \otimes F(\psi )andψ⊗F(φ)\psi \otimes F( \varphi )are isotropic. We consider the case wheredim⁡φ=2n\dim \varphi =2^nandφ\varphiis divisible by an(n−2)(n-2)-fold Pfister form. We determine those formsψ\psifor whichφ\varphibecomes isotropic overF(ψ)F(\psi )ifn≤3n\leq 3, and provide partial results forn≥4n\geq 4. These results imply that ifF(φ)F( \varphi )andF(ψ)F(\psi )are equivalent anddim⁡φ=dim⁡ψ\dim \varphi =\dim \psi, thenφ\varphiis similar toψ\psioverFF. This together with already known results yields that ifφ\varphiis of height22and degree11or22, and ifdim⁡φ=dim⁡ψ\dim \varphi =\dim \psi, thenF(φ)F( \varphi )andF(ψ)F(\psi )are equivalent iffF(φ)F( \varphi )andF(ψ)F(\psi )are isomorphic overFF.
Finite-dimensional lattice-subspaces of 𝐶(Ω) and curves of ℝⁿ
Authors: Polyrakis Ioannis
Published: July 2002

Let

x
1

,
…
,

x
n

x_1,\dotsc ,x_n

be linearly independent positive functions in

C
(
Ω
)

C(\Omega )

, let

X
X

be the vector subspace generated by the

x
i

x_i

and let

β
\beta

denote the curve of

R

n

\mathbb R^n

determined by the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta left-parenthesis t right-parenthesis equals StartFraction 1 Over z left-parenthesis t right-parenthesis EndFraction left-parenthesis x 1 left-parenthesis t right-parenthesis comma x 2 left-parenthesis t right-parenthesis comma ellipsis comma x Subscript n Baseline left-parenthesis t right-parenthesis right-parenthesis">

β
(
t
)
=

1

z
(
t
)

(

x
1

(
t
)
,

x
2

(
t
)
,
…
,

x
n

(
t
)
)

\beta (t)=\frac {1}{z(t)} (x_1(t),x_2(t),\dotsc ,x_n(t))

, where

z
(
t
)
=

x
1

(
t
)
+

x
2

(
t
)
+
⋯
+

x
n

(
t
)

z(t)=x_1(t)+x_2(t)+\dotsb +x_n(t)

. We establish that

X
X

is a vector lattice under the induced ordering from

C
(
Ω
)

C(\Omega )

if and only if there exists a convex polygon of

R

n

\mathbb R^n

with

n
n

vertices containing the curve

β
\beta

and having its vertices in the closure of the range of

β
\beta

. We also present an algorithm which determines whether or not

X
X

is a vector lattice and in case

X
X

is a vector lattice it constructs a positive basis of

X
X

. The results are also shown to be valid for general normed vector lattices.
Wiener’s test for space-time random walks and its applications
Authors: Fukai Yasunari , Uchiyama Kôhei
Published: July 2002

This paper establishes a criterion for whether a

d
d

-dimensional random walk on the integer lattice

Z

d

\mathbf {Z}^{d}

visits a space-time subset infinitely often or not. It is a precise analogue of Wiener’s test for regularity of a boundary point with respect to the classical Dirichlet problem. The test obtained is applied to strengthen the harder half of Kolmogorov’s test for the random walk.
𝐶*-Algebras with the Approximate Positive Factorization Property
Authors: Murphy G. , Phillips N.
Published: July 2002

We say that a unital

C

∗

\mathrm {C}^{*}

-algebra

A
A

has the approximate positive factorization property (APFP) if every element of

A
A

is a norm limit of products of positive elements of

A
A

. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of

C

∗

\mathrm {C}^{*}

-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial

K

1

K_{1}

, and stable rank 1. In the unital case, the

K

0

K_{0}

group separates the tracial states. The APFP passes to matrix algebras, and if

I
I

is an ideal in

A
A

such that

I
I

and

A

/

I

A/I

have the APFP, then so does

A
A

. We also give some new examples of

C

∗

\mathrm {C}^{*}

-algebras with the APFP, including type

I
I

1

\mathrm {II}_{1}

factors and infinite-dimensional simple unital direct limits of homogeneous

C

∗

\mathrm {C}^{*}

-algebras with slow dimension growth, real rank zero, and trivial

K

1

K_{1}

group. Simple direct limits of homogeneous

C

∗

\mathrm {C}^{*}

-algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a

C

∗

\mathrm {C}^{*}

-algebra that may be of interest in the future.
Gross spaces
Authors: Shelah Saharon , Spinas Otmar
Published: July 2002

AGross spaceis a vector spaceEEof infinite dimension over some fieldFF, which is endowed with a symmetric bilinear formΦ:E2→F\Phi :E^{2} \rightarrow Fand has the property that every infinite dimensional subspaceU⊆EU\subseteq Esatisfies dim U⊥>U^{\perp }>dim EE. Gross spaces over uncountable fields exist (in certain dimensions) (see [H. Gross and E. Ogg,Quadratic spaces with few isometries, Comment. Math. Helv. 48 (1973), 511-519]). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariantbequalsω1\omega _{1}, a Gross space in dimensionω1\omega _{1}exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.
Half De Rham complexes and line fields on odd-dimensional manifolds
Authors: Fan Houhong
Published: July 2002

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.
On the ordering of 𝑛-modal cycles
Authors: Bernhardt Chris
Published: July 2002

The forcing relation on

n
n

-modal cycles is studied. If

α
\alpha

is an

n
n

-modal cycle then the

n
n

-modal cycles with block structure that force

α
\alpha

form a

2
n

2^n

-horseshoe above

α
\alpha

. If

n
n

-modal

β
\beta

forces

α
\alpha

, and

β
\beta

does not have a block structure over

α
\alpha

, then

β
\beta

forces a

2
2

-horseshoe of simple extensions of

α
\alpha

.
Exact controllability and stabilizability of the Korteweg-de Vries equation
Authors: Russell David , Zhang Bing-Yu
Published: July 2002

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis i right-parenthesis EndLabel partial-differential Subscript t Baseline u plus u partial-differential Subscript x Baseline u plus partial-differential Subscript x Superscript 3 Baseline u equals f EndLayout">

(i)

∂
t

u
+
u

∂
x

u
+

∂
x
3

u
=
f

\begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*}

on the interval

0
≤
x
≤
2
π
,

t
≥
0

0\leq x\leq 2\pi , \, t\geq 0

, with periodic boundary conditions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis ii right-parenthesis EndLabel partial-differential Subscript x Superscript k Baseline u left-parenthesis 2 pi comma t right-parenthesis equals partial-differential Subscript x Superscript k Baseline u left-parenthesis 0 comma t right-parenthesis comma k equals 0 comma 1 comma 2 comma EndLayout">

(ii)

∂
x
k

u
(
2
π
,
t
)
=

∂
x
k

u
(
0
,
t
)
,

k
=
0
,
1
,
2
,

\begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*}

where the distributed control

f
≡
f
(
x
,
t
)

f\equiv f(x,t)

is restricted so that the “volume”

∫
0

2
π

u
(
x
,
t
)
d
x

\int ^{2\pi }_0 u(x,t) dx

of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control

f
f

is allowed to act on the whole spatial domain

(
0
,
2
π
)

(0,2\pi )

, it is shown that the system is globally exactly controllable, i.e., for given

T
>
0

T> 0

and functions

ϕ
(
x
)

\phi (x)

,

ψ
(
x
)

\psi (x)

with the same “volume”, one can alway find a control

f
f

so that the system (i)–(ii) has a solution

u
(
x
,
t
)

u(x,t)

satisfying
\[

u
(
x
,
0
)
=
ϕ
(
x
)
,

u
(
x
,
T
)
=
ψ
(
x
)
.

u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) .

\]
If the control

f
f

is allowed to act on only a small subset of the domain

(
0
,
2
π
)

(0,2\pi )

, then the same result still holds if the initial and terminal states,

ψ
\psi

and

ϕ
\phi

, have small “amplitude” in a certain sense. In the case of closed loop control, the distributed control

f
f

is assumed to be generated by a linear feedback law conserving the “volume” while monotonically reducing

∫
0

2
π

u
(
x
,
t

)
2

d
x

\int ^{2\pi }_0 u(x,t)^2 dx

. The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.
On a parabolic equation with a singular lower order term
Authors: Zhang Qi
Published: July 2002

We obtain the existence of the weak Green’s functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green’s function, we prove a Harnack inequality for the non-negative weak solutions.
Hyponormality and spectra of Toeplitz operators
Authors: Farenick Douglas , Lee Woo
Published: July 2002

This paper concerns algebraic and spectral properties of Toeplitz operators

T

φ

T_{\varphi }

, on the Hardy space

H

2

(

T

)

H^{2}({\mathbb {T}})

, under certain assumptions concerning the symbols

φ
∈

L

∞

(

T

)

\varphi \in L^{\infty }({\mathbb {T}})

. Among our algebraic results is a characterisation of normal Toeplitz opertors with polynomial symbols, and a characterisation of hyponormal Toeplitz operators with polynomial symbols of a prescribed form. The results on the spectrum are as follows. It is shown that by restricting the spectrum, a set-valued function, to the set of all Toeplitz operators, the spectrum is continuous at

T

φ

T_{\varphi }

, for each quasicontinuous

φ
\varphi

. Secondly, we examine under what conditions a classic theorem of H. Weyl, which has extensions to hyponormal and Toeplitz operators, holds for all analytic functions of a single Toeplitz operator with continuous symbol.
Topological entropy of standard type monotone twist maps
Authors: Knill Oliver
Published: July 2002

We study invariant measures of families of monotone twist maps

S

γ

(
q
,
p
)

S_{\gamma }(q,p)

=
=

(
2
q
−
p
+
γ
⋅

V
′

(
q
)
,
q
)

(2q-p+ \gamma \cdot V’(q),q)

with periodic Morse potential

V
V

. We prove that there exist a constant

C
=
C
(
V
)

C=C(V)

such that the topological entropy satisfies

h

t
o
p

(

S

γ

)
≥
log
⁡
(
C
⋅
γ
)

/

3

h_{top}(S_{\gamma }) \geq \log (C \cdot \gamma )/3

. In particular,

h

t
o
p

(

S

γ

)
→
∞

h_{top}(S_{\gamma }) \to \infty

for

|

γ

|

→
∞

|\gamma | \to \infty

. We show also that there exist arbitrary large

γ
\gamma

such that

S

γ

S_{\gamma }

has nonuniformly hyperbolic invariant measures

μ

γ

\mu _{\gamma }

with positive metric entropy. For large

γ
\gamma

, the measures

μ

γ

\mu _{\gamma }

are hyperbolic and, for a class of potentials which includes

V
(
q
)
=
sin
⁡
(
q
)

V(q)=\sin (q)

, the Lyapunov exponent of the map

S
S

with invariant measure

μ

γ

\mu _{\gamma }

grows monotonically with

γ
\gamma

.
Infinite products of finite simple groups
Authors: Saxl Jan , Shelah Saharon , Thomas Simon
Published: July 2002

We classify the sequences

⟨

S

n

∣
n
∈

N

⟩

\langle S_{n} \mid n \in \mathbb {N} \rangle

of finite simple nonabelian groups such that

∏

n

S

n

\prod _{n} S_{n}

has uncountable cofinality.
On extension of cocycles to normalizer elements, outer conjugacy, and related problems
Authors: Danilenko Alexandre , Golodets Valentin
Published: July 2002

Let

T
T

be an ergodic automorphism of a Lebesgue space and

α
\alpha

a cocycle of

T
T

with values in an Abelian locally compact group

G
G

. An automorphism

θ
\theta

from the normalizer

N
[
T
]

N[T]

of the full group

[
T
]

[T]

is said to be compatible with

α
\alpha

if there is a measurable function

φ
:
X
→
G

\varphi : X \to G

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha left-parenthesis theta x comma theta upper T theta Superscript negative 1 Baseline right-parenthesis equals minus phi left-parenthesis x right-parenthesis plus alpha left-parenthesis x comma upper T right-parenthesis plus phi left-parenthesis upper T x right-parenthesis">

α
(
θ
x
,
θ
T

θ

−
1

)
=
−
φ
(
x
)
+
α
(
x
,
T
)
+
φ
(
T
x
)

\alpha (\theta x, \theta T\theta ^{-1}) = - \varphi (x) + \alpha (x, T) + \varphi (Tx)

at a.e.

x
x

. The topology on the set

D
(
T
,
α
)

D(T, \alpha )

of all automorphisms compatible with

α
\alpha

is introduced in such a way that

D
(
T
,
α
)

D(T , \alpha )

becomes a Polish group. A complete system of invariants for the

α
\alpha

-outer conjugacy (i.e. the conjugacy in the quotient group

D
(
T
,
α
)

/

[
T
]
)

D(T, \alpha )/[T])

is found. Structure of the cocycles compatible with every element of

N
[
T
]

N[T]

is described.
K-theoretic classification for certain inductive limit 𝑍₂ actions on real rank zero 𝐶*-algebras
Authors: Su Hongbing
Published: July 2002

In this paper a K-theoretic classification is given of the C

∗

^*

-algebra dynamical systems

(
A
,
α
,

Z
2

)
=

lim
→

(

A
n

,

α

n

,

Z
2

)

(A, \alpha , Z_2)= \lim \limits _\to (A_n, {\alpha }_n, Z_2)

where

A
A

is of real rank zero, each

A
n

A_n

is a finite direct sum of matrix algebras over finite connected graphs, and each

α
n

\alpha _n

is induced by an action on each component of the spectrum of

A
n

A_n

. Corresponding to the trivial actions is the K-theoretic classification for real rank zero C

∗

^*

-algebras that can be expressed as finite direct sums of matrix algebras over finite graphs obtained in Mem. Amer. Math. Soc. no. 547, vol. 114.
Integral type linear functionals on ordered cones
Authors: Roth Walter
Published: July 2002

We introduce linear functionals on an ordered cone that are minimal with respect to a given subcone. Using concepts developed for Choquet theory we observe that the properties of these functionals resemble those of positive Radon measures on locally compact spaces. Other applications include monotone functionals on cones of convex sets, H-integrals on H-cones in abstract potential theory, and classical Choquet theory itself.
Integer translation of meromorphic functions
Authors: Kim Jeong , Rubel Lee
Published: July 2002

Let

G
G

be a given open set in the complex plane. We prove that there is an entire function such that its integer translations forms a normal family in a neighborhood of

z
z

exactly for

z
z

in

G
G

if and only if

G
G

is periodic with period 1, i.e.,

z
±
1
∈
G

z\pm 1\in G

for all

z
∈
G

z\in G

.
Bloch constants of bounded symmetric domains
Authors: Zhang Genkai
Published: July 2002

Let

D

1

D_{1}

and

D

2

D_{2}

be two irreducible bounded symmetric domains in the complex spaces

V

1

V_{1}

and

V

2

V_{2}

respectively. Let

E
E

be the Euclidean metric on

V

2

V_{2}

and

h
h

the Bergman metric on

V

1

V_{1}

. The Bloch constant

b
(

D

1

,

D

2

)

b(D_{1}, D_{2})

is defined to be the supremum of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis f prime left-parenthesis z right-parenthesis x comma f prime left-parenthesis z right-parenthesis x right-parenthesis Superscript one half slash h Subscript z Baseline left-parenthesis x comma x right-parenthesis Superscript 1 slash 2">

E
(

f

′

(
z
)
x
,

f

′

(
z
)
x

)

1
2

/

h

z

(
x
,
x

)

1

/

2

E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}

, taken over all the holomorphic functions

f
:

D

1

→

D

2

f: D_{1}\to D_{2}

and

z
∈

D

1

z\in D_{1}

, and nonzero vectors

x
∈

V

1

x\in V_{1}

. We find the constants for all the irreducible bounded symmetric domains

D

1

D_{1}

and

D

2

D_{2}

. As a special case we answer an open question of Cohen and Colonna.
Group actions on arrangements of linear subspaces and applications to configuration spaces
Authors: Sundaram Sheila , Welker Volkmar
Published: July 2002

For an arrangement of linear subspaces in

R

n

\mathbb {R}^n

that is invariant under a finite subgroup of the general linear group

G

l
n

(

R

)

Gl_n(\mathbb {R})

we develop a formula for the

G
G

-module structure of the cohomology of the complement

M

A

\mathcal {M}_{\mathcal {A}}

. Our formula specializes to the well known Goresky-MacPherson theorem in case

G
=
1

G = 1

, but for

G
≠
1

G \neq 1

the formula shows that the

G
G

-module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space

M

A

/

G

\mathcal {M}_{\mathcal {A}}/G

. Our motivating examples are arrangements in

C

n

\mathbb {C}^n

that are invariant under the action of

S
n

S_n

by permuting coordinates. A particular case is the “

k
k

-equal” arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases

M

A

\mathcal {M}_{\mathcal {A}}

and

M

A

/

S
n

\mathcal {M}_{\mathcal {A}}/S_n

are spaces of ordered and unordered point configurations in

C

n

\mathbb {C}^n

many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in

R

d

\mathbb {R}^d

and provide explicit results for the “

k
k

-equal” and the “

k
k

-divisible” cases.
Jet Cohomology of Isolated Hypersurface Singularities and Spectral Sequences
Authors: Jian Xiao
Published: July 2002

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra.
Subgroups of finite soluble groups inducing the same permutation character
Authors: Gavioli Norberto
Published: July 2002

In this paper there are found necessary and sufficient conditions that a pair of solvable finite groups, say

G
G

and

K
K

, must satisfy for the existence of a solvable finite group

L
L

containing two isomorphic copies of

G
G

and

H
H

inducing the same permutation character. Also a construction of

L
L

is given as an iterated wreath product, with respect to their actions on their natural modules, of finite one-dimensional affine groups.
Every semigroup is isomorphic to a transitive semigroup of binary relations
Authors: McKenzie Ralph , Schein Boris
Published: July 2002

Every (finite) semigroup is isomorphic to a transitive semigroup of binary relations (on a finite set).
Limiting subhessians, limiting subjets and their calculus
Authors: Ioffe Alexander , Penot Jean-Paul
Published: July 2002

We study calculus rules for limiting subjets of order two. These subjets are obtained as limits of sequences of subjets, a subjet of a function

f
f

at some point

x
x

being the Taylor expansion of a twice differentiable function which minorizes

f
f

and coincides with

f
f

at

x
x

. These calculus rules are deduced from approximate (or fuzzy) calculus rules for subjets of order two. In turn, these rules are consequences of delicate results of Crandall-Ishii-Lions. We point out the similarities and the differences with the case of first order limiting subdifferentials.
Partial regularity of solutions to a class of degenerate systems
Authors: Xu Xiangsheng
Published: July 2002

We consider the system

∂
u

∂
t

−
Δ
u
=
σ

(
u
)

|
∇
φ
|

2

\displaystyle \frac {\partial u }{\partial t}-\Delta u=\sigma \left ( u\right ) \left | \nabla \varphi \right | ^2

,

d
i
v

(
σ

(
u
)

∇
φ
)

=
0

\mathrm {div} \left ( \sigma \left ( u\right ) \nabla \varphi \right ) =0

in

Q
T

≡
Ω
×

(
0
,
T
]

Q_T\equiv \Omega \times \left ( 0,T\right ]

coupled with suitable initial-boundary conditions, where

Ω
\Omega

is a bounded domain in

R

N

\mathbf {R}^N

with smooth boundary and

σ

(
u
)

\sigma \left ( u\right )

is a continuous and positive function of

u
u

. Our main result is that under some conditions on

σ
\sigma

there exists a relatively open subset

Q
0

Q_0

of

Q
T

Q_T

such that

u
u

is locally Hölder continuous on

Q
0

Q_0

, the interior of

Q
T

∖

Q
0

Q_T\backslash Q_0

is empty, and

u
u

is essentially bounded on

Q
T

∖

Q
0

Q_T\backslash Q_0

.
Ramanujan’s class invariants, Kronecker’s limit formula, and modular equations
Authors: Berndt Bruce , Chan Heng Huat , Zhang Liang-Cheng
Published: July 2002

In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan’s class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker’s limit formula, the second employs modular equations, and the third uses class field theory to make Watson’s “empirical method”rigorous.
Congruences between Modular Forms, Cyclic Isogenies of Modular Elliptic Curves, and Integrality of 𝑝-adic 𝐿-Functions
Authors: Tang Shu-Leung
Published: July 2002

Let

Γ
\Gamma

be a congruence subgroup of type

(

N
1

,

N
2

)

(N_1,N_2)

and of level

N
N

. We study congruences between weight 2 normalized newforms

f
f

and Eisenstein series

E
E

on

Γ
\Gamma

modulo a prime

p

\mathfrak {p}

above a rational prime

p
p

. Assume that

p
∤
6
N

p\nmid 6N

,

E
E

is a common eigenfunction for all Hecke operators and

f
f

is ordinary at

p

\mathfrak {p}

. We show that the abelian variety associated to

f
f

and the cuspidal subgroup associated to

E
E

intersect non-trivially in their

p
p

-torsion points. Let

A
A

be a modular elliptic curve over

Q

\mathbb {Q}

with good ordinary reduction at

p
p

. We apply the above result to show that an isogeny of degree divisible by

p
p

from the optimal curve

A
1

A_1

in the

Q

\mathbb {Q}

-isogeny class of elliptic curves containing

A
A

to

A
A

extends to an étale morphism of Néron models over

Z

p

\mathbb {Z}_p

if

p
>
7

p>7

. We use this to show that

p
p

-adic distributions associated to the

p
p

-adic

L
L

-functions of

A
A

are

Z

p

\mathbb {Z}_p

-valued.
Automorphism Groups and Invariant Subspace Lattices
Authors: Muhly Paul , Solel Baruch
Published: July 2002

Let

(
B
,

R

,
α
)

(B,\mathbf {R},\alpha )

be a

C

∗

C^{*}

- dynamical system and let

A
=

B
α

(
[
0
,
∞
)
)

A=B^\alpha ([0,\infty ))

be the analytic subalgebra of

B
B

. We extend the work of Loebl and the first author that relates the invariant subspace structure of

π
(
A
)
,

\pi (A),

for a

C

∗

C^{*}

-representation

π
\pi

on a Hilbert space

H

π

\mathcal {H}_\pi

, to the possibility of implementing

α
\alpha

on

H

π

.

\mathcal {H}_\pi .

We show that if

π
\pi

is irreducible and if lat

π
(
A
)

\pi (A)

is trivial, then

π
(
A
)

\pi (A)

is ultraweakly dense in

L
(
H

π

)
.

\mathcal {L(H}_\pi ).

We show, too, that if

A
A

satisfies what we call the strong Dirichlet condition, then the ultraweak closure of

π
(
A
)

\pi (A)

is a nest algebra for each irreducible representation

π
.

\pi .

Our methods give a new proof of a “density” theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid

C

∗

C^{*}

-algebras.
Hecke algebras, 𝑈_{𝑞}𝑠𝑙_{𝑛}, and the Donald-Flanigan conjecture for 𝑆_{𝑛}
Authors: Gerstenhaber Murray , Schaps Mary
Published: July 2002

The Donald–Flanigan conjecture asserts that the integral group ring

Z

G

\mathbb {Z}G

of a finite group

G
G

can be deformed to an algebra

A
A

over the power series ring

Z

[
[
t
]
]

\mathbb {Z}[[t]]

with underlying module

Z

G
[
[
t
]
]

\mathbb {Z}G[[t]]

such that if

p
p

is any prime dividing

#
G

\#G

then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A circled-times Subscript double-struck upper Z left-bracket left-bracket t right-bracket right-bracket Baseline ModifyingAbove double-struck upper F Subscript p Baseline left-parenthesis left-parenthesis t right-parenthesis right-parenthesis With bar">

A

⊗

Z

[
[
t
]
]

F

p

(
(
t
)
)

¯

A\otimes _{\mathbb {Z}[[t]]}\overline {\mathbb {F}_{p}((t))}

is a direct sum of total matric algebras whose blocks are in natural bijection with and of the same dimensions as those of

C

G
.

\mathbb {C}G.

We prove this for

G
=

S

n

G = S_{n}

using the natural representation of its Hecke algebra

H

\mathcal {H}

by quantum Yang-Baxter matrices to show that over

Z

[
q
]

\mathbb {Z}[q]

localized at the multiplicatively closed set generated by

q
q

and all
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i Subscript q squared Baseline equals 1 plus q squared plus q Superscript 4 Baseline plus midline-horizontal-ellipsis plus q Superscript 2 left-parenthesis i minus 1 right-parenthesis Baseline comma i equals 1 comma 2 comma ellipsis comma n">

i

q

2

=
1
+

q

2

+

q

4

+
⋯
+

q

2
(
i
−
1
)

,
i
=
1
,
2
,
…
,
n

i_{q^{2}} = 1+q^{2} + q^{4} + \dots + q^{2(i-1)}, i = 1,2,\dots , n

, the Hecke algebra becomes a direct sum of total matric algebras. The corresponding “canonical" primitive idempotents are distinct from Wenzl’s and in the classical case (

q
=
1

q=1

), from those of Young.
On the Kolyvagin cup product
Authors: Besser Amnon
Published: July 2002

We define a new cohomological operation, which we call the Kolyvagin cup product, that is a generalization of the derivative operator introduced by Kolyvagin in his work on Euler systems. We show some of the basic properties of this operation. We also define a higher dimensional derivative in certain cases and a dual operation which we call the Kolyvagin cap product and which generalizes a computation of Rubin.
Reversion of power series and the extended Raney coefficients
Authors: Cheng Charles , McKay James , Towber Jacob , Wang Stuart , Wright David
Published: July 2002

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients.
A bound on the geometric genus of projective varieties verifying certain flag conditions
Authors: Di Gennaro Vincenzo
Published: July 2002

Fix integersn,r,s1,...,sln,r,s_{1},...,s_{l}and letS(n,r;s1,...,sl)\mathcal {S}(n,r;s_{1},...,s_{l})be the set of all integral, projective and nondegenerate varietiesVVof degrees1s_{1}and dimensionnnin the projective spacePr\mathbf {P}^{r}, such that, for alli=2,...,li=2,...,l,VVdoes not lie on any variety of dimensionn+i−1n+i-1and degree>si>s_{i}. We say that a varietyVVsatisfies aflag conditionof type(n,r;s1,...,sl)(n,r;s_{1},...,s_{l})ifVVbelongs toS(n,r;s1,...,sl)\mathcal {S}(n,r;s_{1},...,s_{l}). In this paper, under the hypothesess1>>...>>sls_{1}>>...>>s_{l}, we determine an upper boundGh(n,r;s1,...,sl)G^{h}(n,r;s_{1},...,s_{l}), depending only onn,r,s1,...,sln,r,s_{1},...,s_{l}, for the number<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G left-parenthesis n comma r semicolon s 1 comma period period period comma s Subscript l Baseline right-parenthesis colon equals m a x StartSet p Subscript g Baseline left-parenthesis upper V right-parenthesis colon upper V element-of script upper S left-parenthesis n comma r semicolon s 1 comma period period period comma s Subscript l Baseline right-parenthesis EndSet">G(n,r;s1,...,sl):=max{pg(V):V∈S(n,r;s1,...,sl)}G(n,r;s_{1},...,s_{l}):= {max} {\{} p_{g}(V) : V\in \mathcal {S}(n,r;s_{1},...,s_{l}){\}}, wherepg(V)p_{g}(V)denotes the geometric genus ofVV. In casen=1n=1andl=2l=2, the study of an upper bound for the geometric genus has a quite long history and, forn≥1n\geq 1,l=2l=2ands2=r−ns_{2}=r-n, it has been introduced by Harris. We exhibit sharp results for particular ranges of our numerical datan,r,s1,...,sln,r,s_{1},...,s_{l}. For instance, we extend Halphen’s theorem for space curves to the case of codimension two and characterize the smooth complete intersections of dimensionnninPn+3\mathbf {P}^{n+3}as the smooth varieties of maximal geometric genus with respect to appropriate flag condition. This result applies to smooth surfaces inP5\mathbf {P}^{5}. Next we discuss how farGh(n,r;s1,...,sl)G^{h}(n,r;s_{1},...,s_{l})is fromG(n,r;s1,...,sl)G(n,r;s_{1},...,s_{l})and show a sort of lifting theorem which states that, at least in certain cases, the varietiesV∈S(n,r;s1,...,sl)V\in \mathcal {S}(n,r;s_{1},...,s_{l})of maximal geometric genusG(n,r;s1,...,sl)G(n,r;s_{1},...,s_{l})must in fact lie on a flag such as<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V equals upper V Subscript s 1 Superscript n Baseline subset-of upper V Subscript s 2 Superscript n plus 1 Baseline subset-of period period period subset-of upper V Subscript s Sub Subscript l Subscript Superscript n plus l minus 1 Baseline subset-of bold upper P Superscript r">V=Vs1n⊂Vs2n+1⊂...⊂Vsln+l−1⊂PrV=V_{s_{1}}^{n}\subset V_{s_{2}}^{n+1}\subset ...\subset V_{s_{l}}^{n+l-1}\subset {\mathbf {P}^{r}}, whereVsjV^{j}_{s}denotes a subvariety ofPr\mathbf {P}^{r}of degreessand dimensionjj. We also discuss further generalizations of flag conditions, and finally we deduce some bounds for Castelnuovo’s regularity of varieties verifying flag conditions.
Tarski’s finite basis problem via 𝐀(𝒯)
Authors: Willard Ross
Published: July 2002

R. McKenzie has recently associated to each Turing machine

T

\mathcal {T}

a finite algebra

A

(

T

)

\mathbf {A} (\mathcal {T})

having some remarkable properties. We add to the list of properties, by proving that the equational theory of

A

(

T

)

\mathbf {A}(\mathcal {T})

is finitely axiomatizable if

T

\mathcal {T}

halts on the empty input. This completes an alternate (and simpler) proof of McKenzie’s negative answer to A. Tarski’s finite basis problem. It also removes the possibility, raised by McKenzie, of using

A

(

T

)

\mathbf {A}(\mathcal {T})

to answer an old question of B. Jónsson.