bluuebunny/crossref_metadata_2025 huggingface dataset:
Extending families of disjoint zero sets
Authors: Aull C. E.
Published: June 2010

The

z
z

-cellularity

z
(
X
)

z(X)

of a space

X
X

is defined as
\[

z
(
X
)
=
sup

{

|
Z
|

:
Z
⊂
Z
(
X
)

}

z(X) = \sup \left \{ {\left | Z \right |:Z \subset Z(X)} \right \}

\]
where

Z
(
X
)

Z(X)

is the family of zero sets of

X
X

. It is proved using CH that a Tychonoff space

S
S

is

T

C
∗

T{C^ * }

-embedded in every Tychonoff space it is

C
C

-embedded in iff

z
(
S
)
⩽
c

z(S) \leqslant c

. A space

S
S

is defined to be

T

C
∗

T{C^ * }

-embedded in a space

X
X

if any disjoint family of zero sets of

S
S

can be extended to a family of disjoint zero sets of

X
X

. Similar theorems are proved for

C
∗

{C^ * }

-embedding when

S
S

is a

P
P

-space or the zero sets have the Isiwata property.
Finitely Boolean representable varieties
Authors: Kiss Emil W.
Published: June 2010

This paper gives a short, elementary proof of a result of Burris and McKenzie [2] stating that each variety Boolean representable by a finite set of finite algebras is the join of an abelian and a discriminator variety. An example showing that the Boolean product operatorΓa{\Gamma ^a}is not idempotent is included as well.
Starshapedness of level sets for the obstacle problem and for the capacitory potential problem
Authors: Kawohl Bernhard
Published: June 2010

If the geometry of the data in the obstacle problem and the capacitory potential problem is starshaped then so are the solutions. The proofs are based on appropriate maximum principles.
Quaternion Kaehlerian manifolds isometrically immersed in Euclidean space
Authors: Barros M. , Urbano F.
Published: June 2010

Let

M
M

be a complete

4
n

4n

-dimensional quaternion Kaehlerian manifold isometrically immersed in the

(
4
n
+
d
)

(4n + d)

-dimensional Euclidean space. In this note we prove that if

d
>
n

d > n

, then

M
M

is a Riemannian product

Q
m

×
P

{Q^m} \times P

, where

Q
m

{Q^m}

is the

m
m

-dimensional quaternion Euclidean space

(
m
⩾
n
−
d
)

(m \geqslant n - d)

and

P
P

is a Ricci flat quaternion Kaehlerian manifold.
Non-Tychonoff 𝑒-compactifiable spaces
Authors: Hart K. P. , Vermeer J.
Published: June 2010

We construct a non-Tychonoff space

X
X

which is

e
e

-compactifiable, thus answering a question of S. Hechler. We also answer a question of R. M. Stephenson: whether there exists a Tychonoff space, the largest

e
e

-compactification of which has a noncompact semiregularization.
Relative Barr-Rinehart and cotriple cohomology groups are isomorphic
Authors: Van Osdol D. H.
Published: June 2010

The theorem, stated in the title of this article, is proved.
Property 𝐶 and fine homotopy equivalences
Authors: van Mill Jan , Mogilski Jerzy
Published: June 2010

We show that within the class of metric

σ
\sigma

-compact spaces, proper fine homotopy equivalences preserve property

C
C

, which is a slight generalization of countable dimensionality. We also give an example of an open fine homotopy equivalence of a countable dimensional space onto a space containing the Hilbert cube.
A generalization of Altman’s ordering principle
Authors: Turinici Mihai
Published: June 2010

A maximality principle on quasi-ordered quasi-metric spaces, containing, in particular, Altman’s and the Brézis-Browder ordering principles, is given. As an application, a local mapping theorem extending—from a "functional" viewpoint—a similar one due to Altman, is derived.
A reflexive space of holomorphic functions in infinitely many variables
Authors: Alencar Raymundo , Aron Richard M. , Dineen Seán
Published: June 2010

We show the existence of an infinite-dimensional Banach space

E
E

such that

H
(
E
)

H(E)

, the space of holomorphic functions on

E
E

, endowed with the

τ
ω

{\tau _\omega }

topology is reflexive.
Two Helly type theorems
Authors: Kołodziejczyk Krzysztof
Published: June 2010

Two Helly type theorems for convex sets containing

k
k

-dimensional half-flats are established, one of them being an extension of Katchalski’s theorem [3] where the case

k
=
1

k = 1

was considered.
Some random fixed point theorems for condensing operators
Authors: Sehgal V. M. , Waters Charles
Published: June 2010

In this paper we obtain several random fixed point theorems including a stochastic generalization of the classical Rothe fixed point theorem. The results herein improve a recent result of Bharucha-Reid and Mukherjea and also some similar results of Itoh.
Finite group actions and nonseparating 2-spheres
Authors: Plotnick Steven P.
Published: June 2010

We extend the splitting theorem of Meeks-Yau for finite group actions on three-manifolds to include manifolds containing nonseparating

2
2

-spheres, and give applications to branched covers of links.
Branched cyclic covers of simple knots
Authors: Strickland Paul
Published: June 2010

The Blanchfield pairing of simple

(

2
q
−
1

)

\left ( {2q - 1} \right )

-knots,

q
⩾
2

q \geqslant 2

, is used to give an algebraic characterization of those knots which may arise as

m
m

-fold branched cyclic covers of simple knots.
An elementary approach to generic properties of plane curves
Authors: Bruce J. W. , Giblin P. J.
Published: June 2010

A direct and geometrical approach to the proof of generic properties of smooth submanifolds of Euclidean space is illustrated in the simplest case of plane curves.
Recovery of 𝐻^{𝑝}-functions
Authors: Totik V.
Published: June 2010

Let there be given finitely many points

{

α
k

}
1
n

\{ {\alpha _k}\} _1^n

from the unit disc. If

f
f

is a

H
p

{H^p}

-function then how well can the value of

f
f

at

z
=
0

z = 0

be approximated by linear means

∑
1
n

c
k

f
(

α
k

)
?

\sum \nolimits _1^n {{c_k}f({\alpha _k})?}

We give the optimal constants

c
k

{c_k}

and get, as a corollary, the possibility of the approximation of

f
f

by operators of the form

∑
1
n

f
(

α
k

)

p
k

\sum \nolimits _1^n {f({\alpha _k}){p_k}}

with polynomials

p
k

{p_k}

. The order of approximation depends on the distance

∑
1
n

(

1
−

|

α
k

|

)

\sum \nolimits _1^n {\left ( {1 - \left | {{\alpha _k}} \right |} \right )}

of the point system from the unit circle.
Universality of small lattice varieties
Authors: Koubek V. , Sichler J.
Published: June 2010

There exists a finitely generated lattice variety

S
S

such that the class of all nonconstant homomorphisms between members of

S
S

contains a universal category as a full subcategory. In particular, every monoid

M
M

is isomorphic to the monoid of all nonconstant endomorphisms of a lattice from

S
S

, and

S
S

contains arbitrarily large lattices representing

M
M

. The category of all

(
0
,
1
)

(0,1)

-homomorphisms of lattices in

S
S

is also shown to be universal.
Maximal and minimal solutions to a class of elliptic quasilinear problems
Authors: Troianiello Giovanni M.
Published: June 2010

We prove existence of maximal and minimal solutions to bilateral problems for quasilinear elliptic operators with nondivergence principal part independent of the gradient. This result also covers the case of equations, when the obstacles can be taken as lower and upper solutions.
On a conjecture of M. S. Robertson
Authors: Lyzzaik Abdallah
Published: June 2010

We prove that two classes of univalent functions are equal. This settles a conjecture of M. S. Robertson in the affirmative.
Nonseparating almost continuous retracts of 𝐼ⁿ
Authors: Rosen Harvey
Published: June 2010

Compact almost continuous retracts of

I
n

(
n
⩾
2
)

{I^n}(n \geqslant 2)

do not separate

E
n

{E^n}

. Some other results that hold for continuous functions are also shown to hold for almost continuous functions. A result in [5] giving sufficient conditions for a set to be an almost continuous retract of

I
n

{I^n}

is examined further, and a method of constructing some almost continuous retracts of

I
n

{I^n}

is given.
A tower of spectra that realizes a chain complex
Authors: Suárez Pedro A.
Published: June 2010

This paper presents the construction of a tower of spectra

Y
j

{Y_j}

with

k
k

-invariants coming from the relations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S q Superscript 1 Baseline left-parenthesis upper X upper S q Superscript 2 Super Superscript j plus 1 Superscript Baseline right-parenthesis upper S q Superscript 1 Baseline left-parenthesis upper X upper S q Superscript 2 Super Superscript j Superscript Baseline right-parenthesis equals 0">

S

q

1

(
X

S

q

2

j
+
1

)

S

q

1

(
X

S

q

2
j

)
=
0

{\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^{j + 1}}}}){\text {S}}{{\text {q}}^1}(X{\text {S}}{{\text {q}}^{{2^j}}}) = 0

in

A

/

A

S

q

1

A / A{\text {S}}{{\text {q}}^1}

, for

0
⩽
j
⩽
5

0 \leqslant j \leqslant 5

and

A
A

= Steenrod algebra

mod

2

\mod 2

, such that

Y
5

{Y_5}

has prescribed homotopy groups:

π
n

(

Y
5

)
=
Z

{\pi _n}({Y_5}) = Z

(integers) if

n
=

2

j
+
1

−
2

n = {2^{j + 1}} - 2

, and zero otherwise.
Relations for Grothendieck groups of Artin algebras
Authors: Auslander Maurice
Published: June 2010

M. C. R. Butler has shown that if

Λ
\Lambda

is an artin algebra of finite representation type, then the almost split sequences generate the relations for the Grothendieck group of

Λ
\Lambda

. This paper is primarily devoted to proving the converse of Butler’s result, i.e.

Λ
\Lambda

is of finite representation type if the almost split sequences generate the relations for the Grothendieck group of

Λ
\Lambda

.
Uniform algebras and projections
Authors: Sidney S. J.
Published: June 2010

If

M
M

is a closed

A
A

-submodule of

C

(
X
)

C\left ( X \right )

where

A
A

is a uniform algebra on

X
X

which contains a separating family of unimodular functions, and if

M
M

is a quotient space of some

C

(
Y
)

C\left ( Y \right )

, then

M
M

is an ideal in

C

(
X
)

C\left ( X \right )

. If there is an example of a uniform algebra

A
A

on some

X
X

such that

A
≠
C

(
X
)

A \ne C\left ( X \right )

but

A
A

is complemented in

C

(
X
)

C\left ( X \right )

, then there is such an example with

A
A

separable.
A coefficient problem of Bombieri concerning univalent functions
Authors: Bshouty Daoud
Published: June 2010

We answer affirmatively a question raised by Bombieri concerning the behaviour of the coefficients of normalized univalent functions near the Koebe function.
The fixed point property for continua approximated from within by Peano continua with this property
Authors: Tominaga Akira
Published: June 2010

Let

X
X

be a continuum that is approximated from within by Peano subcontinua with the fixed point property (FPP). Then we show a sufficient condition that

X
X

has FPP. As a consequence we have that the Cartesian product of

n
n

Warsaw circles is a

T
n

{T^n}

-like continuum with FPP and the

n
n

-fold suspension of Warsaw circle is an

S
n

{S^n}

-like continuum with FPP.
A note on Ralph Fox’s 𝛾-space
Authors: Künzi Hans-Peter A.
Published: June 2010

We show that Ralph Fox’s construction of a non-quasi-metrizable

γ
\gamma

-space gives an example of a preorthocompact space that has an open cover that is not quasi-normal.
Multidimensional analytic structure and Shilov boundaries
Authors: Kumagai Donna
Published: June 2010

We give a condition under which multi-dimensional analytic structure can be introduced into the maximal ideal space of a uniform algebra.
Regularity of the distance function
Authors: Foote Robert L.
Published: June 2010

A coordinate-free proof is given of the fact that the distance function

δ
\delta

for a

C
k

{C^k}

submanifold

M
M

of

R

n

{{\mathbf {R}}^n}

is

C
k

{C^k}

near

M
M

when

k
⩾
2

k \geqslant 2

. The result holds also when

k
=
1

k = 1

if

M
M

has a neighborhood with the unique nearest point property. The differentiability of

δ
\delta

in the

C
1

{C^1}

case is seen to follow directly from geometric considerations.
Invariants related to the Bergman kernel of a bounded domain in 𝐶ⁿ
Authors: Kanemaru Tadayoshi
Published: June 2010

In this paper we introduce biholomorphic invariants using the Bergman kernel function of a bounded domain in

C

n

{{\mathbf {C}}^n}

.
Multiple points of a random field
Authors: Shieh Narn Rueih
Published: June 2010

We prove that a

d
d

-dimensional random field

X
≡

{
X
(
t
)

}

t
∈

R
+
N

X \equiv {\{ X(t)\} _{t \in R_ + ^N}}

has uncountably many

r
r

-multiple points a.s. when it satisfies Pitt’s (

(

A
r

)

({A_r})

) condition [9]. Those

t
t

’s for which

X
(
t
)

X(t)

hits the multiple point can be separated by any given positive number, and multiple points can occur while

t
t

is restricted to any given "time inteval". Two corollaries concerning Gaussian fields and fields with independent increments are also presented.
Maximal 𝑝-locals
Authors: Robinson Geoffrey R.
Published: June 2010

We prove a theorem on maximal

p
p

-local subgroups of finite groups which has the Baer-Suzuki theorem as a corollary.
Differential basis and 𝑝-basis of a regular local ring
Authors: Kimura Tetsuzo , Niitsuma Hiroshi
Published: June 2010

We prove that any differential basis of a regular local ring

R
R

of characteristic

p
>
0

p > 0

over

R
p

{R^p}

is a

p
p

-basis of

R
R

over

R
p

{R^p}

. This result gives a characterization of a regular local ring

R
R

which has a

p
p

-basis over

R
p

{R^p}

.
On existence of oscillatory solutions for a second order sublinear differential equation
Authors: Wong James S. W.
Published: June 2010

A result on the existence of oscillatory solutions for the second order sublinear differential equation

y
+
a
(
t
)

|
y
|

γ

sgn
⁡
y
=
0

y + a(t){\left | y \right |^\gamma }\operatorname {sgn} y = 0

,

0
>
γ
>
1

0 > \gamma > 1

, where

a
(
t
)

a(t)

is positive and continuous, is given. This supplements a well-known result of Hinton for the superlinear case, i.e.

γ
>
1

\gamma > 1

.
Approximation theorems for zero-sum nonstationary stochastic games
Authors: Nowak Andrzej S.
Published: June 2010

This paper deals with zero-sum nonstationary stochastic games with countable state and action spaces which include both Shapley’s stochastic games [11] and infinite games with imperfect information studied by Orkin in [7]. It is shown that any nonstationary stochastic game with a bounded below lower semicontinuous payoff defined on the space of all histories has a value function and the minimizer has an optimal strategy. Moreover, two approximation theorems extending the main results of Orkin from [7] are established. Finally, counterexamples answering in the negative some open questions raised by Orkin [7] and Sengupta [10] are given.
Products of closed images of CW-complexes and 𝑘-spaces
Authors: Tanaka Yoshio , Zhou Hao Xuan
Published: June 2010

We give some characterizations for the products of closed images of CW-complexes to be

k
k

-spaces.
The power structure of metabelian 𝑝-groups
Authors: Blackburn Norman , Espuelas Alberto
Published: June 2010

In a metabelian

p
p

-group in which the

p
p

th powers generate a subgroup of order

p
p

, the elements of order

p
p

generate a subgroup of index at most

p
p

{p^p}

. This is best possible.
Ergodic and mixing properties of measures on locally compact abelian groups
Authors: Ramsey Thomas , Weit Yitzhak
Published: June 2010

We provide new proofs of the theorems of Choquet and Deny and of Foguel concerning iterates of convolutions of a measure on a locally compact abelian group.
Holomorphic separation and the union problem
Authors: Vigna Suria Giuseppe
Published: June 2010

We give a sufficient condition for an increasing union of holomorphically separated analytic spaces to be holomorphically separated. Furthermore, an example of J. E. Fornaess is investigated in order to show that a union as above is not always holomorphically separated.
On Halpern’s conjecture for closed plane curves
Authors: Ozawa Tetsuya
Published: June 2010

Let

c
c

be a smooth closed plane curve given in general position. A bitangent of

c
c

is, by definition, a line which is tangent to

c
c

at two different points. Let

B
(
c
)

B(c)

and

D
(
c
)

D(c)

denote the numbers of all bitangents and all double points of

c
c

, respectively. We prove here that if

c
c

has no inflection points,

B
(
c
)
⩽
D
(
c
)
(
2
D
(
c
)
−
1
)

B(c) \leqslant D(c)(2D(c) - 1)

. This is the affirmative answer to Halpern’s conjecture.
On some cyclotomic congruences of F. Thaine
Authors: Washington Lawrence C.
Published: June 2010

Some congruences involving cyclotomic integers, orginally obtained by F. Thaine, are proved and strengthened by using

p
p

-adic logarithms and

p
p

-adic

L
L

-functions.
A formula for an equivalent invariant measure
Authors: Maharam Dorothy
Published: June 2010

Let

T
T

be a nonsingular

1
−
1

1 - 1

measurable transformation of a finite measure space

(
X
,
B
,
m
)

(X,B,m)

. A simple construction is given for a

σ
\sigma

-finite measure

μ
\mu

, equivalent to

m
m

and invariant under

T
T

, when such a measure exists.
On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational
Authors: McKenna P. J.
Published: June 2010

The semilinear wave equation

u

t
t

−

u

x
x

+
f
(
u
)
=
g
(
t
)

{u_{tt}} - {u_{xx}} + f(u) = g(t)

with

u
(
0
,
t
)
=
u
(
π
,
t
)
=
0

u(0,t) = u(\pi ,t) = 0

,

u
u

is

T
T

-periodic in

t
t

, is considered for some situations in which

T
T

is not a rational multiple of

π
\pi

. Various existence results depending on the range of

f
′

f’

are given, which contrast sharply with the case where

T
T

is a rational multiple of

π
\pi

.
A rigid space whose square is the Hilbert space
Authors: Dijkstra Jan J.
Published: June 2010

We construct a space

X
X

with only one autohomeomorphism and the property that the product

X
×
X

X \times X

is homeomorphic to the separable Hilbert space.
Fake boundary sets in the Hilbert cube
Authors: Bowers Philip L.
Published: June 2010

For each positive integer

n
n

, a

σ
\sigma

-

Z
Z

-set

B
n

{B_n}

in the Hilbert cube

I
∞

{I^\infty }

is constructed whose complement

s
n

=

I
∞

−

B
n

{s_n} = {I^\infty } - {B_n}

is not homeomorphic to the pseudointerior

s
s

of the Hilbert cube though

s
n

{s_n}

and

B
n

{B_n}

satisfy: (i) every compact subset of

s
n

{s_n}

is a

Z
Z

-set in

s
n

{s_n}

; (ii)

s
n

×

s
n

{s_n} \times {s_n}

is homeomorphic to

s
s

; (iii)

B
n

{B_n}

admits small maps

I
∞

→

B
n

{I^\infty } \to {B_n}

; (iv)

s
n

{s_n}

satisfies the discrete

n
n

-cells property; and (v)

B
n

{B_n}

is locally

(
n
−
1
)

(n - 1)

-connected in

I
∞

{I^\infty }

. It is shown that

s
n

{s_n}

does not satisfy the discrete

(
n
+
1
)

(n + 1)

-cells property and thus

B
n

{B_n}

is not a boundary set, that is,

s
n

{s_n}

is not homeomorphic to

s
s

. These examples build upon an example of Anderson, Curtis, and van Mill of a fake boundary set

B
0

{B_0}

that satisfies (i)-(iv) for

n
=
0

n = 0

. Their example is not a boundary set since it fails to be locally continuum-connected. The examples constructed herein show that there is a hierarchy of fake boundary sets satisfying (i)-(iv) that satisfy higher and higher orders of a strong form of local connectivity (v).
Insertion, approximation, and extension of real-valued functions
Authors: Blair Robert L. , Swardson Mary Anne
Published: June 2010

For a uniformly closed vector lattice

V
V

of real-valued functions on a set

X
X

, necessary and sufficient conditions are obtained for insertion (or "strict insertion") of some member of

V
V

between two arbitrary real-valued functions on

X
X

. These conditions quickly yield known insertion, approximation, and extension theorems for real-valued functions.
Remark on the class number of 𝑄(√2𝑝) modulo 8 for 𝑝≡5(𝑚𝑜𝑑8) a prime
Authors: Williams Kenneth S. , Friesen Christian
Published: June 2010

An explicit congruence modulo 8 is given for the class number of the real quadratic field

Q
(

2
p

)

Q(\sqrt {2p} )

, where

p
p

is a prime congruent to 5 modulo 8.
Generalized intersection multiplicities of modules. II
Authors: Dutta Sankar P.
Published: June 2010

The vanishing conjecture for intersection multiplicities in dimensions

⩽
5

\leqslant 5

is true in greater generality than previously known.
Local uncertainty inequalities for Fourier series
Authors: Price John F. , Racki Paul C.
Published: June 2010

Necessary and sufficient conditions are given on

α
\alpha

,

β
\beta

and

t
t

for there to exist a constant

K
K

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis sigma-summation Underscript n element-of upper E Endscripts StartAbsoluteValue ModifyingAbove f With caret left-parenthesis n right-parenthesis EndAbsoluteValue squared right-parenthesis Superscript 1 slash 2 Baseline less-than-or-slanted-equals upper K StartAbsoluteValue upper E EndAbsoluteValue Superscript alpha Baseline double-vertical-bar f StartAbsoluteValue x EndAbsoluteValue Superscript beta Baseline double-vertical-bar Subscript t">

(

∑

n
∈
E

|

f
^

(
n
)

|

2

)

1

/

2

⩽
K

|
E
|

α

‖

f

|
x
|

β

‖

t

{\left ( {{{\sum \limits _{n \in E} {\left | {\hat f(n)} \right |} }^2}} \right )^{1/2}} \leqslant K{\left | E \right |^\alpha }{\left \| {f{{\left | x \right |}^\beta }} \right \|_t}

\]
for all

f
∈

L
1

(

T
d

)

f \in {L^1}({T^d})

and finite

E
⊂

Z
d

E \subset {Z^d}

.
The radial behavior of Blaschke products
Authors: Belna C. L. , Colwell P. , Piranian G.
Published: June 2010

The paper extends results of G. T. Cargo and of C. L. Belna, F. W. Carroll and G. Piranian on the existence of Blaschke products with prescribed radial limits at a prescribed finite or countable set of points on the unit circle. The extension permits the prescription of the radial cluster set at each point of a prescribed countable set on the unit circle.
On the spectral multiplicity of a class of finite rank transformations
Authors: Goodson G. R.
Published: June 2010

The rank

M
M

transformations, which Chacon called the simple approximations with multiplicity

M
M

, were shown by Chacon to have maximal spectral multiplicity at most

M
M

, although no example was given where this bound is attained for

M
>
1

M > 1

. In this paper, for each natural number

M
>
1

M > 1

, we show how to construct a simple approximation with multiplicity

M
M

which is ergodic and has maximal spectral multiplicity equal to

M
−
1

M - 1

.
Maximal compact normal subgroups and pro-Lie groups
Authors: Bagley R. W. , Wu T. S.
Published: June 2010

We are concerned with conditions under which a locally compact group

G
G

has a maximal compact normal subgroup

K
K

and whether or not

G

/

K

G/K

is a Lie group. If

G
G

has small compact normal subgroups

K
K

such that

G

/

K

G/K

is a Lie group, then

G
G

is pro-Lie. If in

G
G

there is a collection of closed normal subgroups

{

H
α

}

\{ {H_\alpha }\}

such that

∩

H
α

=
e

\cap {H_\alpha } = e

and

G

/

H
α

G/{H_\alpha }

is a Lie group for each

α
\alpha

, then

G
G

is a residual Lie group. We determine conditions under which a residual Lie group is pro-Lie and give an example of a residual Lie group which is not embeddable in a pro-Lie group.
Nonuniqueness of the cardinality attached to homogeneous 𝐴𝑊*-algebras
Authors: Ozawa Masanao
Published: June 2010

It is proved that for any pair of infinite cardinal numbers

α
\alpha

and

β
\beta

, there exists a homogeneous AW*-algebra that is both

α
\alpha

-homogenous and

β
\beta

-homogeneous. This negatively settles a long-standing unsolved problem of type I AW*-algebras.
Simply connected spin manifolds with positive scalar curvature
Authors: Miyazaki Tetsuro
Published: June 2010

Let

l
n

{l_n}

be equal to 2 or 4 according as

n
n

is congruent

mod

8

\mod 8

to zero or not. If the

A
^

\hat A

-genus of a compact simply connected spin manifold of dimension

n
⩾
5

n \geqslant 5

vanishes, then the connected sum of

l
n

{l_n}

copies of the manifold admits a metric of positive scalar curvature. This supports a conjecture of Gromov and Lawson.
Homology classes which are represented by graph links
Authors: Yano Koichi
Published: June 2010

We give a necessary and sufficient condition for a one-dimensional homology class of a graph manifold to be represented by a graph link.
Nil derivations and chain conditions in prime rings
Authors: Chung L. O. , Kobayashi Y.
Published: June 2010

It is known that in a prime ring, a derivation nilpotent on a nonzero ideal must also be nilpotent in the entire ring. In this paper we show that a derivation of a prime ring is not necessarily nil even though it is nil on a nonzero ideal. It is nil if the ring satisfies the ascending chain condition on left (right) annihilator ideals.
On Zimmermann-Huisgen’s splitting theorem
Authors: Camillo Victor
Published: June 2010

This note is motivated by a paper of Birge Zimmermann-Huisgen, which in turn is motivated by a long sequence of papers—the first due to Faith—dealing with the question of when the canonical embedding of a direct sum of modules in the corresponding direct product splits. Zimmermann-Huisgen answered a question raised by previous authors by showing that if

R
R

is a von Neumann regular ring the only way this can happen is that, except for a finite number, the modules involved must each be semisimple with only a finite number of simple modules involved. Based on a new, more elementary argument, we establish a necessary condition for the sum-product splitting over an arbitrary (associative) ring ft (with identity).
Remarks on a paper by A. Aziz
Authors: Rubinstein Zalman
Published: June 2010

The paper consists of two parts. In the first part a short proof of the main theorem due to A. Aziz on the location of zeros of composite polynomials is given. In the second part some properties of a fixed length continuation of a polynomial are deduced.
Sturmian theorems for second order systems
Authors: Allegretto W.
Published: June 2010

Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l 1 ModifyingAbove u With right-arrow equals minus normal upper Delta ModifyingAbove u With right-arrow plus upper A ModifyingAbove u With right-arrow comma l 2 ModifyingAbove w With right-arrow equals minus normal upper Delta ModifyingAbove w With right-arrow plus upper B ModifyingAbove w With right-arrow">

l
1

u
→

=
−
Δ

u
→

+
A

u
→

,

l
2

w
→

=
−
Δ

w
→

+
B

w
→

{l_1}\vec u = - \Delta \vec u + A\vec u,{l_2}\vec w = - \Delta \vec w + B\vec w

, and Dirichlet boundary conditions. Here

Δ
\Delta

denotes the Laplace operator, and

A
,
B

A,B

are

m
×
m

m \times m

matrices. We do not assume that

A
,
B

A,B

are symmetric, but instead essentially require

B
B

irreducible and

b

i
j

⩽
0

 if 

i
≠
j

{b_{ij}} \leqslant 0{\text { if }}i \ne j

. Estimates on the real eigenvalue of

l
2

{l_2}

, with a positive eigenvector are then obtained as applications. Our results are motivated by recent theorems for ordinary differential equations established by Ahmad, Lazer and Dannan.
On Lipschitz functions of normal operators
Authors: Kittaneh Fuad
Published: June 2010

It is shown that if

N
N

and

M
M

are normal operators on a separable, complex Hilbert space

H
H

, and

f
f

is a Lipschitz function on

Ω
=
σ
(
N
)
∪
σ
(
M
)

\Omega = \sigma (N) \cup \sigma (M)

(i.e.,

|

f
(
z
)
−
f
(
w
)

|

⩽
k

|

z
−
w

|

\left | {f(z) - f(w)} \right | \leqslant k\left | {z - w} \right |

for some positive constant

k
k

and all

z
,
w
∈
Ω
)

z,w \in \Omega )

, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f left-parenthesis upper N right-parenthesis upper X minus upper X f left-parenthesis upper M right-parenthesis double-vertical-bar Subscript 2 Baseline less-than-or-slanted-equals k double-vertical-bar upper N upper X minus upper X upper M double-vertical-bar Subscript 2">

‖

f
(
N
)
X
−
X
f
(
M
)

‖

2

⩽
k

‖

N
X
−
X
M

‖

2

{\left \| {f(N)X - Xf(M)} \right \|_2} \leqslant k{\left \| {NX - XM} \right \|_2}

for any operator

X
X

on

H
H

. In particular,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f left-parenthesis upper N right-parenthesis minus f left-parenthesis upper M right-parenthesis double-vertical-bar Subscript 2 Baseline less-than-or-slanted-equals k double-vertical-bar upper N minus upper M double-vertical-bar Subscript 2">

‖

f
(
N
)
−
f
(
M
)

‖

2

⩽
k

‖

N
−
M

‖

2

{\left \| {f(N) - f(M)} \right \|_2} \leqslant k{\left \| {N - M} \right \|_2}

.
Analyticity in the boundary of a pseudoconvex domain
Authors: Noell Alan V.
Published: June 2010

Let

D
D

be a bounded pseudoconvex domain with

C
∞

{C^\infty }

boundary in

C

n

,

A
∞

(
D
)

{{\mathbf {C}}^n},{A^\infty }(D)

the algebra of functions holomorphic in

D
D

and

C
∞

{C^\infty }

up to the boundary, and

M
M

a compact real-analytic manifold in the boundary which is integral for the complex structure of the boundary and which has no complex tangent vectors. A necessary and sufficient condition that each element of

A
∞

(
D
)

{A^\infty }(D)

be real-analytic on

M
M

is that the germ of the complexification of

M
M

be in the boundary. Examples indicate that the quasi-analyticity of

A
∞

(
D
)

{A^\infty }(D)

along

M
M

is possible even in the absence of complex manifolds in the boundary.
A lattice of conditions on topological spaces
Authors: Collins P. J. , Reed G. M. , Roscoe A. W. , Rudin M. E.
Published: June 2010

If

W
(
x
)

W(x)

(for each

x
∈
X

x \in X

) is a family of subsets each containing

x
x

, various conditions on

{

W
(
x
)
:
x
∈
X

}

\left \{ {W(x):x \in X} \right \}

are investigated. They yield new criteria for paracompactness, metrisability and the existence of a semimetric generating a given topology.
Topological spaces with point-networks
Authors: Balogh Zoltán T.
Published: June 2010

In this paper we introduce the notion of a point-network and prove that in

T
1

{T_1}

spaces having a point-network, countable pseudocharacter implies stratifiable. As a corollary, there follows Reed’s theorem that a

T
1

{T_1}

space is metrizable if and only if it has an open point-network.
A note on the Bott vanishing theorem
Authors: Moskowitz Ira
Published: June 2010

We give a simple example of a foliated

4
4

-manifold that shows that the bound in Bott’s theorem is best possible.
A formal analogue of Hilbert’s Theorem 90
Authors: Coleman R.
Published: June 2010

We prove a theorem on

p
p

-adic analytic functions which formally resembles Hilbert’s Theorem 90 and which solves a problem originally proposed by Deligne and considered by Adolphson [A].
Isolated chain recurrent points for one-dimensional maps
Authors: Block Louis , Franke John E.
Published: June 2010

For a continuous map of the interval to itself or of the circle to itself, we show that any isolated chain recurrent point is eventually periodic. Furthermore, an isolated chain recurrent point which is not in the orbit of a critical point and has no critical point in its orbit is periodic.
The unique representation of a selfadjoint bounded linear functional
Authors: Suen Ching Yun
Published: June 2010

It is well known that every selfadjoint bounded linear functional on a

C
∗

{C^ * }

-algebra has a unique minimal decomposition [6, Theorem 3.2.5]. In this paper we prove that under some conditions a selfadjoint completely bounded linear map with a unique minimal decomposition is equivalent to the map with a unique commutant representation (up to unitary equivalence). Using the results, we generalize the Gel’fand-Naimark-Segal construction.
A perfectly normal, locally compact, noncollectionwise normal space from ♢*
Authors: Daniels Peg , Gruenhage Gary
Published: June 2010

A perfectly normal, locally compact, collectionwise-

T
2

{T_2}

, noncollectionwise normal space is constructed using

♢
∗

{\diamondsuit ^*}

, a combinatorial axiom which holds in Gödel’s constructible universe

L
L

. The construction answers questions of

F

{\text {F}}

. Tall and S. Watson.
On the algebraic characteristic set for a class of matroids
Authors: Lindström Bernt
Published: June 2010

The independent sets of an algebraic matroid are sets of algebraically independent transcendentals over a field

k
k

. If a matroid

M
M

is isomorphic to an algebraic matroid the latter is called an algebraic representation of

M
M

. Vector representations of matroids are defined similarly. A matroid may have algebraic (resp. vector) representations over fields of different characteristics. The problem in which characteristic sets are possible for vector representations was recently answered (see [2]). The corresponding problem for algebraic representations is open. We consider a class of matroids

M
p

{M_p}

(

p
p

a prime) the vector representations which were determined by T. Lazarson long ago. One member of this class,

M
2

{M_2}

, is the important Fano matroid which plays a crucial role in many parts of matroid theory. We prove that

M
p

{M_p}

has algebraic representations only over fields of characteristic

p
p

. The proof depends on derivations in fields. Using derivations we transform an algebraic representation of

M
p

{M_p}

into a vector representation.
Bipowers in number fields
Authors: Harrison D. K.
Published: June 2010

The set of all solutions to the Fermat equation is given a structure. This structure is then characterized up to isomorphism in terms of certain subsets of the integers modulo a prime.
Singular solutions of the heat equation with absorption
Authors: Kamin S. , Peletier L. A.
Published: June 2010

In this paper we prove that the source-type solutions converge— when the total initial mass tends to infinity—to the very singular solution obtained in [3].
Note on 𝐻² on planar domains
Authors: Bañuelos R. , Wolff T.
Published: June 2010

An analytic function

f
:
U
∪

U
′

→

C

f:U \cup U’ \to {\mathbf {C}}

,

U
U

,

U
′

⊂

C

U’ \subset {\mathbf {C}}

may belong to

H
2

(
U
)

{H^2}(U)

and

H
2

(

U
′

)

{H^2}(U’)

and not to

H
2

(
U
∪

U
′

)

{H^2}(U \cup U’)

.
Explicit examples of Bloch functions in every 𝐻^{𝑝} space, but not in BMOA
Authors: Holland Finbarr , Twomey J. Brian
Published: June 2010

It is shown how to construct analytic functions, with nonnegative Taylor coefficients, that belong to the intersection of the space of Bloch functions and all the

H
p

{H^p}

spaces and yet do not have bounded mean oscillation.
Perturbations and ground states of 𝐶*-dynamical systems
Authors: Kurose Hideki
Published: June 2010

In this paper we show that if a

C
∗

{C^ * }

-dynamical system has an irreducible covariant representation, every relatively bounded

∗
*

-derivation for its generator is also implemented by a relatively bounded selfadjoint operator for that associated with the dynamics. As its application, we assert that the existence of ground states of a

C
∗

{C^ * }

-dynamical system is stable under sufficiently small perturbations.
Conformal invariants of Minkowski space
Authors: Morava Jack
Published: June 2010

The conformal invariant defined for compact Riemannian manifolds by Yamabe is generalized to pseudo-Riemannian manifolds and is shown to be nontrivial for Minkowski space. We also make some elementary remarks about generalizations of Yamabe’s equation to sections of vector bundles, as have been studied by physicists concerned with Goldstone bosons and the Higgs mechanism.
A tool in establishing total variation convergence
Authors: Parthasarathy K. R. , Steerneman Ton
Published: June 2010

Let

X
0

,

X
1

,

X
2

,
…

 and 

Y
0

,

Y
1

,

Y
2

,
…

{X_0},{X_1},{X_2}, \ldots {\text { and }}{Y_0},{Y_1},{Y_2}, \ldots

be sequences of random variables where

X
n

{X_n}

and

Y
n

{Y_n}

are independent,

L

X
n

→
L

X
0

L{X_n} \to L{X_0}

in total variation and

L

Y
n

→
L

Y
0

L{Y_n} \to L{Y_0}

in distribution. For certain mappings

T
T

sufficient conditions are given in order that

L
T

(

X
n

,

Y
n

)

→
L
T

(

X
0

,

Y
0

)

LT\left ( {{X_n},{Y_n}} \right ) \to LT\left ( {{X_0},{Y_0}} \right )

in total variation. For example, if

(

R

k

,

B
k

)

\left ( {{{\mathbf {R}}^k},{B_k}} \right )

is the outcome space of the

X
n

{X_n}

and

Y
n

{Y_n}

, and if

L

X
0

L{X_0}

is absolutely continuous (with respect to Lebesgue measure), then

L

(

X
n

+

Y
n

)

→
L

(

X
0

+

Y
0

)

L\left ( {{X_n} + {Y_n}} \right ) \to L\left ( {{X_0} + {Y_0}} \right )

in total variation.
Scattered compactifications and the orderability of scattered spaces. II
Authors: Purisch S.
Published: June 2010

A space is suborderable if it is embeddable in a (totally) orderable space. It is shown that a suborderable scattered space is orderable and admits an orderable scattered compactification.
Elliptic operators and a theorem of Poincaré
Authors: Karp Leon
Published: June 2010

A general vanishing theorem is proved for elliptic operators. This result is then used to give a simple proof of the fact that the arithmetic genus vanishes for complex manifolds of odd dimension

n
n

with nowhere zero

(

n
,
0

)

\left ( {n,0} \right )

form.
On the structure of 𝑉𝑁(𝐺) for almost connected groups
Authors: Mashhood B.
Published: June 2010

We prove that for any almost connected locally compact group

G
G

, the von Neumann algebra

VN

(
G
)

{\text {VN}}\left ( G \right )

, generated by the left regular representation of

G
G

is semifinite.
Polynomial hulls with convex sections and interpolating spaces
Authors: Slodkowski Zbigniew
Published: June 2010

Assume that

L
⊂
∂
D
×

C

m

L \subset \partial D \times {{\mathbf {C}}^m}

is compact and has convex vertical sections. Denote by

K
K

its polynomially convex hull. It is shown that

K
∖
∂
D
×

C

m

K\backslash \partial D \times {{\mathbf {C}}^m}

, if nonempty, can be covered by graphs of analytic functions

f
:
D
→

C

m

f:D \to {{\mathbf {C}}^m}

. The proof is based on complex interpolation theory for families of finite-dimensional normed spaces.
On a factorization problem for convergent sequences and on Hankel forms in bounded sequences
Authors: Eggermont P. P. B. , Leung Y. J.
Published: June 2010

We solve in the negative the following factorization problem of S. Mazur: Can every convergent sequence be written as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="z left-parenthesis n right-parenthesis equals left-parenthesis n plus 1 right-parenthesis Superscript negative 1 Baseline sigma-summation Underscript i equals 0 Overscript n Endscripts x left-parenthesis i right-parenthesis y left-parenthesis n minus i right-parenthesis comma n equals 0 comma 1 comma ellipsis">

z
(
n
)
=

(
n
+
1

)

−
1

∑

i
=
0

n

x
(
i
)
y
(
n
−
i
)
,
n
=
0
,
1
,
…

z(n) = {(n + 1)^{ - 1}}\sum \nolimits _{i = 0}^n {x(i)y(n - i),n = 0,1, \ldots }

, with convergent sequences

x
x

and

y
y

? This problem also yields the solution of another problem of S. Mazur regarding bounded Hankel forms on the space of all bounded sequences.
A completeness theorem for trigonometric identities and various results on exponential functions
Authors: van den Dries Lou
Published: June 2010

All valid identities in terms of variables, real constants, the arithmetic operations of addition and multiplication, and the trigonometric operations of sine and cosine are shown to be consequences of a few familiar identities and numerical facts. We also indicate how to decide whether

f
f

eventually dominates

g
g

, for

f
f

and

g
g

from a certain class of exponential functions. Finally, we correct a statement from an earlier paper.
Analytic functionals and the Bergman projection on circular domains
Authors: Zorn Paul
Published: June 2010

A property of the Bergman projection associated to a bounded circular domain containing the origin in

C

N

{{\mathbf {C}}^N}

is proved: Functions which extend to be holomorphic in large neighborhoods of the origin are characterized as Bergman projections of smooth functions with small support near the origin. For certain circular domains

D
D

, it is also shown that functions which extend holomorphically to a neighborhood of

D
¯

\overline D

are precisely the Bergman projections of smooth functions whose supports are compact subsets of

D
D

. Two applications to proper holomorphic mappings are given.
A singular integral
Authors: Namazi Javad
Published: June 2010

In this paper we show that if

K
(
x
)
=
Ω
(
x
)

/

|
x
|

n

K(x) = \Omega (x)/{\left | x \right |^n}

is a Calderón-Zygmund kernel, where

Ω
∈

L
q

(

S

n
−
1

)

\Omega \in {L^q}({S^{n - 1}})

for some

1
>
q
≤
∞

1 > q \leq \infty

, and

b
b

is a radial bounded function, then

b
(
x
)
K
(
x
)

b(x)K(x)

is the kernel of a convolution operator which is bounded on

L
p

(

R
n

)

{L^p}({R^n})

for

1
>
p
>
∞

1 > p > \infty

and

n
≥
2

n \geq 2

.
Uniformly persistent systems
Authors: Butler Geoffrey , Freedman H. I. , Waltman Paul
Published: June 2010

Conditions are given under which weak persistence of a dynamical system with respect to the boundary of a given set implies uniform persistence.
On approximation by rationals from a hyperplane
Authors: Gierz Gerhard , Shekhtman Boris
Published: June 2010

Let

E
⊂
C
(
K
)

E \subset C(K)

be a subspace of continuous functions defined on a compact Hausdorff space

K
K

. We characterize those subspaces of codimension 1 for which the rational functions with denominators and enumerators from

E
E

are dense. The condition for the density of this very nonlinear set of functions turns out to be a linear separation condition.
An operator approach to the principle of inclusion and exclusion
Authors: Liu C. J.
Published: June 2010

Using an operator approach we derive Sylvester-Whitworth formulae for sets

A
A

’s. By the same token we treat the problem where both sets of

A
A

’s and

B
B

’s are involved. Our result extends the Sylvester-Whitworth inclusion and exclusion formula to the resolution of the number of elements in exactly

m
1

{m_1}

sets of

A
A

’s and

m
2

{m_2}

sets of

B
B

’s respectively. The formula are applied to the complete graph and complete bipartite graph. The enumeration of spanning subgraphs with any preassigned number of disconnected cycles is solved, together with the case where any preassigned number of vertices have degree one.
Torsion in the cohomology of line bundles on homogeneous spaces for Chevalley groups
Authors: Andersen Henning Haahr
Published: June 2010

In this paper we study the torsion submodules of the cohomology modules for line bundles on

G

/

B

G/B

where

G
G

is a Chevalley group over

Z

{\mathbf {Z}}

and

B
B

a Borel subgroup.
Properties of endomorphism rings of modules and their duals
Authors: Khuri Soumaya Makdissi
Published: June 2010

Let

R

M

_RM

be a nonsingular left

R
R

-module whose Morita context is nondegenerate, let

B
=

End

R

⁡
M

B = \operatorname {End}_{R}M

and let

M
∗

=

Hom

R

⁡
(
M
,
R
)

{M^ * } = \operatorname {Hom}_{R}(M,R)

. We show that

B
B

is left (right) strongly modular if and only if any element of

B
B

which has zero kernel in

R

M
(

M
R
∗

)

_RM(M_R^ * )

has essential image in

R

M
(

M
R
∗

)

_RM(M_R^ * )

, and that

B
B

is a left (right) Utumi ring if and only if every submodule

R

U

o

f

R

M
(

U
R
∗

of 

M
R
∗

)

_RU{\text {o}}{{\text {f}}_R}M(U_R^ * {\text {of }}M_R^ * )

such that

U
⊥

=
0

(
⊥

U
∗

=
0
)

{U^ \bot } = 0{(^ \bot }{U^ * } = 0)

is essential in

R

M
(

M
R
∗

)

_RM(M_R^ * )

.
Riemann step function approximation of Bochner integrable functions
Authors: Freedman M. A.
Published: June 2010

Let

L
1

(
0
,
T
;
X
)

{L^1}(0,T;X)

denote the space of all Bochner integrable functions

f
f

which map the interval

[
0
,
T
]

[0,T]

into the Banach space

X
X

. Then we show that

f
f

is the uniform limit in the

L
1

{L^1}

-norm of its Riemann step function approximations along nearly every sequence of partitions of

[
0
,
T
]

[0,T]

with mesh size approaching zero.
Loop spaces of finite complexes at large primes
Authors: McGibbon C. A. , Wilkerson C. W.
Published: June 2010

Let

X
X

be a finite, simply connected CW complex with only a finite number of nonzero rational homotopy groups. Localized away from a certain finite set of primes, the loop space of

X
X

is shown to be homotopy equivalent to a product of spheres and loop spaces of spheres. Applications to the homotopy groups of

X
X

and the homological properties of

Ω
X

\Omega X

are given.
Existence results on the one-dimensional Dirichlet problem suggested by the piecewise linear case
Authors: Arias M.
Published: June 2010

We study the existence of solutions of a two-point boundary value problem at resonance in which the nonlinearity grows at most linearly. Sharp results for the linear growth of the nonlinearity in each direction are obtained.
Notes on smooth function germs on varieties
Authors: Izumiya Shyūichi , Matsuoka Sachiko
Published: June 2010

We study smooth function germs on varieties from the point of view of Thom-Mather theory.
Points de continuité d’une fonction séparément continue
Authors: Debs Gabriel
Published: June 2010

Let

f
:
X
×
Y
→

R

f:X \times Y \to {\mathbf {R}}

be a separately continuous function defined on the product of a Baire space

X
X

and a Hausdorff compact space

Y
Y

. We prove that

f
f

is jointly continuous at any point of

G
×
Y

G \times Y

, for a dense

G
δ

{G_\delta }

subset

G
G

of

X
X

, under one of the following assumptions: (i) If the Banach space

C

(
Y
)

\mathcal {C}(Y)

is weakly

K

\mathcal {K}

-analytic; or (ii) if

X
X

contains a dense subset which is

K

\mathcal {K}

-analytic.
A group theoretical equivalent of the zero divisor problem
Authors: Lichtman A. I.
Published: June 2010

We prove that the zero divisor problem is equivalent to some problem about subgroups of a free group.
Non-Artinian essential extensions of simple modules
Authors: Goodearl K. R. , Schofield A. H.
Published: June 2010

A noetherian ring of Krull dimension one is constructed which possesses a simple module with a cyclic non-artinian essential extension. The construction also provides an example of a torsionfree noetherian bimodule which fails to satisfy Jategaonkar’s density condition.
A new inequality for complex-valued polynomial functions
Authors: Rassias Themistocles M.
Published: June 2010

Let

f
1

,

f
2

,
…
,

f
n

:

C

→

C

{f_1},{f_2}, \ldots ,{f_n}:{\mathbf {C}} \to {\mathbf {C}}

be complex-valued polynomial functions of degrees

d
1

,

d
2

,
…
,

d
n

{d_1},{d_2}, \ldots ,{d_n}

, respectively, of a complex variable

z
z

. Then
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M Subscript f 1 Baseline upper M Subscript f 2 Baseline midline-horizontal-ellipsis upper M Subscript f Sub Subscript n Baseline greater-than-or-equal-to upper M Subscript f 1 f 2 midline-horizontal-ellipsis f Sub Subscript n Baseline greater-than-or-equal-to k upper M Subscript f 1 Baseline upper M Subscript f 2 Baseline midline-horizontal-ellipsis upper M Subscript f Sub Subscript n">

M

f
1

M

f
2

⋯

M

f
n

≥

M

f
1

f
2

⋯

f
n

≥
k

M

f
1

M

f
2

⋯

M

f
n

{M_{{f_1}}}{M_{{f_2}}} \cdots {M_{{f_n}}} \geq {M_{{f_1}{f_2} \cdots {f_n}}} \geq k{M_{{f_1}}}{M_{{f_2}}} \cdots {M_{{f_n}}}

\]
where
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k equals left-parenthesis sine StartFraction 2 Over n EndFraction StartFraction pi Over 8 d 1 EndFraction right-parenthesis Superscript d 1 Baseline left-parenthesis sine StartFraction 2 Over n EndFraction StartFraction pi Over 8 d 2 EndFraction right-parenthesis Superscript d 2 Baseline midline-horizontal-ellipsis left-parenthesis sine StartFraction 2 Over n EndFraction StartFraction pi Over 8 d Subscript n Baseline EndFraction right-parenthesis Superscript d Super Subscript n Superscript Baseline period">

k
=

(

sin
⁡

2
n

π

8

d
1

)

d
1

(

sin
⁡

2
n

π

8

d
2

)

d
2

⋯

(

sin
⁡

2
n

π

8

d
n

)

d
n

.

k = {\left ( {\sin \frac {2}{n}\frac {\pi } {{8{d_1}}}} \right )^{{d_1}}}{\left ( {\sin \frac {2} {n}\frac {\pi }{{8{d_2}}}} \right )^{{d_2}}} \cdots {\left ( {\sin \frac {2}{n}\frac {\pi }{{8{d_n}}}} \right )^{{d_n}}}.

\]
Picard and Brauer groups of Zariski schemes
Authors: Blass Piotr , Hoobler Raymond
Published: June 2010

The Cartier-Yuan exact sequence is used to calculate Picard groups and Brauer groups of Zariski surfaces and their generalizations. A result of Blass-Deligne on the factoriality of general affine Zariski surfaces is extended to all higher dimensional Zariski schemes.