bluuebunny/crossref_metadata_2025 huggingface dataset:
Pettis integrability
Authors: Stefánsson Gunnar F.
Published: June 2012

A weakly measurable function

f
:
Ω
→
X

f:\Omega \to X

is said to be determined by a subspace

D
D

of

X
X

if for each

x

∗

∈

X

∗

{x^{\ast } } \in {X^{\ast } }

,

x

∗

|

D

=
0

{x^{\ast }}{|_D} = 0

implies that

x

∗

f
=
0

{x^{\ast }}\;f= 0

a.e. For a given Dunford integrable function

f
:
Ω
→
X

f:\Omega \to X

with a countably additive indefinite integral we show that

f
f

is Pettis integrable if and only if

f
f

is determined by a weakly compactly generated subspace of

X
X

if and only if

f
f

is determined by a subspace which has Mazur’s property. We show that if

f
:
Ω
→
X

f:\Omega \to X

is Pettis integrable then there exists a sequence (

φ
n

{\varphi _n}

) of

X
X

valued simple functions such that for all

x

∗

∈

X

∗

{x^{\ast }} \in {X^{\ast }}

,

x

∗

f
=

lim
n

x

∗

φ
n

{x^{\ast }}f= {\lim _n}{x^{\ast }}\,{\varphi _n}

a.e. if and only if

f
f

is determined by a separable subspace of

X
X

. For a bounded weakly measurable function

f
:
Ω
→

X

∗

f:\Omega \to {X^{\ast } }

into a dual of a weakly compactly generated space, we show that

f
f

is Pettis integrable if and only if

f
f

is determined by a separable subspace of

X

∗

{X^{\ast }}

if and only if

f
f

is weakly equivalent to a Pettis integrable function that takes its range in

cor

f

∗

(
Ω
)

{\text {cor}}_f^{\ast } (\Omega )

.
The minimal degree of a finite inverse semigroup
Authors: Schein Boris M.
Published: June 2012

The minimal degree of an inverse semigroup

S
S

is the minimal cardinality of a set

A
A

such that

S
S

is isomorphic to an inverse semigroup of one-to-one partial transformations of

A
A

. The main result is a formula that expresses the minimal degree of a finite inverse semigroup

S
S

in terms of certain subgroups and the ordered structure of

S
S

. In fact, a representation of

S
S

by one-to-one partial transformations of the smallest possible set

A
A

is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries.
Brauer-Hilbertian fields
Authors: Fein Burton , Saltman David J. , Schacher Murray
Published: June 2012

Let

F
F

be a field of characteristic

p
p

(

p
=
0

p = 0

allowed), and let

F
(
t
)

F(t)

be the rational function field in one variable over

F
F

. We say

F
F

is Brauer-Hilbertian if the following holds. For every

α
\alpha

in the Brauer group

Br
⁡
(
F
(
t
)
)

\operatorname {Br}(F(t))

of exponent prime to

p
p

, there are infinitely many specializations

t
→
a
∈
F

t \to a \in F

such that the specialization

α
¯

∈
Br
⁡
(
F
)

\bar \alpha \in \operatorname {Br}(F)

is defined and has exponent equal to that of

α
\alpha

. We show every global field is Brauer-Hilbertian, and if

K
K

is Hilbertian and

F
F

is finite separable over

K
(
t
)

K(t)

,

F
F

is Brauer-Hilbertian.
Stable-homotopy and homology invariants of boundary links
Authors: Farber Michael
Published: June 2012

An

n
n

-dimensional

(
n
≥
5
)

(n \geq 5)

link in the

(
n
+
2
)

(n + 2)

-dimensional sphere is stable if the

i
i

th homotopy group of its complement

X
X

vanishes for

2
≤
i
≤
(
n
+
1
)

/

3

2 \leq i \leq (n + 1)/3

and

π
1

(
X
)

{\pi _1}(X)

is freely generated by meridians. In this paper a classification of stable links in terms of stable homotopy theory is given. For simple links this classification gives a complete algebraic description. We also study Poincaré duality in the space of the free covering of the complement of a boundary link. The explicit computation of the corresponding Ext-functors gives a construction of new homology pairings, generalizing the Blanchfield and the torsion pairings for knots.
Equivariant cohomology and lower bounds for chromatic numbers
Authors: Kříž Igor
Published: June 2012

We introduce a general topological method for obtaining a lower bound of the chromatic number of an

n
n

-graph. We present numerical lower bounds for intersection

n
n

-graphs.
A PL geometric study of algebraic 𝐾 theory
Authors: Hu Bi Zhong
Published: June 2012

This paper manages to apply the Farrell-Jones theory on algebraic

K
K

-groups of closed negatively curved riemannian manifolds to Gromov’s hyperbolic group theory. The paper reaches the conclusion that for any finite polyhedron

K
K

with negative curvature,

Wh
⁡
(

π
1

K
)
=
0

\operatorname {Wh}({\pi _1}K) = 0

.
Examples of pseudo-Anosov homeomorphisms
Authors: Bauer Max
Published: June 2012

We generalize a construction in knot theory to construct a large family

G

R

=
∪

G
R
(

P

)

\mathcal {G}\mathcal {R} = \cup \,GR(\mathcal {P})

of mapping classes of a surface of genus

g
g

and one boundary component, where

P

\mathcal {P}

runs over some finite index set. We exhibit explicitly the set

G

R

∗

⊂

G

R

\mathcal {G}{\mathcal {R}^{\ast }} \subset \mathcal {G}\mathcal {R}

that consists of pseudo-Anosov maps, find the map that realizes the smallest dilatation in

G

R

∗

\mathcal {G}{\mathcal {R}^{\ast }}

, and for every

P

\mathcal {P}

, we give a set of defining relations for

G
R
(

P

)

GR(\mathcal {P})

.
Rigidity of 𝑝-completed classifying spaces of alternating groups and classical groups over a finite field
Authors: Ishiguro Kenshi
Published: June 2012

A

p
p

-adic rigid structure of the classifying spaces of certain finite groups

π
\pi

, including alternating groups

A
n

{A_n}

and finite classical groups, is shown in terms of the maps into the

p
p

-completed classifying spaces of compact Lie groups. The spaces

(
B
π

)
p
∧

(B\pi )_p^ \wedge

have no nontrivial retracts. As an application, it is shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper B upper A Subscript n Baseline right-parenthesis Subscript p Superscript logical-and Baseline asymptotically-equals left-parenthesis upper B normal upper Sigma Subscript n Baseline right-parenthesis Subscript p Superscript logical-and">

(
B

A
n

)
p
∧

≃
(
B

Σ
n

)
p
∧

(B{A_n})_p^ \wedge \simeq (B{\Sigma _n})_p^ \wedge

if and only if

n
≢
0
,
1
,

mod

p

n\not \equiv 0,1,\;\bmod \,p

. It is also shown that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper B upper S upper L left-parenthesis n comma double-struck upper F Subscript q Baseline right-parenthesis right-parenthesis Subscript p Superscript logical-and Baseline asymptotically-equals left-parenthesis upper B upper G upper L left-parenthesis n comma double-struck upper F Subscript q Baseline right-parenthesis right-parenthesis Subscript p Superscript logical-and">

(
B
S
L
(
n
,

F

q

)

)
p
∧

≃
(
B
G
L
(
n
,

F

q

)

)
p
∧

(BSL(n,{\mathbb {F}_q}))_p^ \wedge \simeq (BGL(n,{\mathbb {F}_q}))_p^ \wedge

where

q
q

is a power of

p
p

if and only if

(
n
,
q
−
1
)
=
1

(n,q - 1) = 1

.
Subgroup conditions for groups acting freely on products of spheres
Authors: Silverman Judith H.
Published: June 2012

Let

d
d

and

h
h

be integers such that either

d
≥
2

d \geq 2

and

h
=

2
d

−
1

h = {2^d} - 1

, or

d
=
4

d = 4

and

h
=
5

h = 5

. Suppose that the group

G

\mathcal {G}

contains an elementary-abelian

2
2

-subgroup

E
d

{E_d}

of rank

d
d

with an element

σ
\sigma

of order

h
h

in its normalizer. We show that if

G

\mathcal {G}

admits a free and

F

2

{{\mathbf {F}}_2}

-cohomologically trivial action on

(

S
n

)
d

{({S^n})^d}

, then some nontrivial power of

σ
\sigma

centralizes

E
d

{E_d}

. The cohomology ring
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript asterisk Baseline left-parenthesis upper E Subscript d Baseline semicolon bold upper F 2 right-parenthesis asymptotically-equals bold upper F 2 left-bracket y 1 comma ellipsis comma y Subscript d Baseline right-bracket">

H

∗

(

E
d

;

F

2

)
≃

F

2

[

y
1

,
…
,

y
d

]

{H^{\ast } }({E_d};{{\mathbf {F}}_2}) \simeq {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]

is a module over the Steenrod algebra

A

(
2
)

\mathcal {A}(2)

. Let

θ
∈

F

2

[

y
1

,
…
,

y
d

]

\theta \in {{\mathbf {F}}_2}[{y_1}, \ldots ,{y_d}]

, and let

c
≥
d
−
2

c \geq d - 2

be an integer. We show that

θ
\theta

divides

S

q

2
i

(
θ
)

S{q^{{2^i}}}(\theta )

in the polynomial ring for

0
≤
i
≤
c
⇔
θ
=

τ

2

c
−
d
+
3

π

0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi

, where

τ
\tau

divides

S

q

2
i

(
τ
)

S{q^{{2^i}}}(\tau )

for

0
≤
i
≤
d
−
3

0 \leq i \leq d - 3

and

π
\pi

is a product of linear forms.
On conjugacy separability of fundamental groups of graphs of groups
Authors: Shirvani M.
Published: June 2012

A complete determination of when the elements of a fundamental group of a (countable) graph of profinite groups are conjugacy distinguished is given. By embedding an arbitrary fundamental group

G
G

into one with profinite vertex groups and making use of the above result, questions on conjugacy separability of

G
G

can be reduced to the solution of equations in the vertex groups of

G
G

.
𝑁-body observables in the Calkin algebra
Authors: Dereziński Jan
Published: June 2012

The commutators of many operators which are used in the phase space analysis of the

N
N

-body scattering are compact. This fact makes it possible to give a description of certain classes of such operators in terms of commutative

C

∗

{C^{\ast } }

-algebras inside the Calkin algebra.
Lannes’ 𝑇 functor on summands of 𝐻*(𝐵(𝑍/𝑝)^{𝑠})
Authors: Harris John C. , Shank R. James
Published: June 2012

Let

H
H

be the

mod
-
p

\bmod \text {-}p

cohomology of the classifying space

B
(

Z

/

p
)

B({\mathbf {Z}}/p)

thought of as an object in the category,

U

\mathcal {U}

, of unstable modules over the Steenrod algebra. Lannes constructed a functor

T
:

U

→

U

T:\mathcal {U} \to \mathcal {U}

which is left adjoint to the functor

A
↦
A
⊗
H

A \mapsto A \otimes H

. In this paper we evaluate

T
T

on the indecomposable

U

\mathcal {U}

-summands of

H

⊗
s

{H^{ \otimes s}}

, the tensor product of

s
s

copies of

H
H

. Our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring

F

p

[

M

s
,

s

(

Z

/

p
)
]

{{\mathbf {F}}_p}[{{\mathbf {M}}_{s,}}_s({\mathbf {Z}}/p)]

. The main application is to determine the homotopy type of the space of maps from

B
(

Z

/

p
)
]

B({\mathbf {Z}}/p)]

to

X
X

when

X
X

is a wedge summand of the space

Σ
(
B

(

Z

/

p

)
s

)

\Sigma (B{({\mathbf {Z}}/p)^s})

.
Multipliers of families of Cauchy-Stieltjes transforms
Authors: Hibschweiler R. A. , MacGregor T. H.
Published: June 2012

For

α
>
0

\alpha > 0

let

F

α

{\mathcal {F}_\alpha }

denote the class of functions defined for

|

z

|

>
1

|z| > 1

by integrating

1

/

(
1
−
x
z

)
α

1/{(1 - xz)^\alpha }

against a complex measure on

|

x

|

=
1

|x|= 1

. A function

g
g

holomorphic in

|

z

|

>
1

|z| > 1

is a multiplier of

F

α

{\mathcal {F}_\alpha }

if

f
∈

F

α

f \in {\mathcal {F}_\alpha }

implies

g
f
∈

F

α

gf \in {\mathcal {F}_\alpha }

. The class of all such multipliers is denoted by

M

α

{\mathcal {M}_\alpha }

. Various properties of

M

α

{\mathcal {M}_\alpha }

are studied in this paper. For example, it is proven that

α
>
β

\alpha > \beta

implies

M

α

⊂

M

β

{\mathcal {M}_\alpha } \subset {\mathcal {M}_\beta }

, and also that

M

α

⊂

H
∞

{\mathcal {M}_\alpha } \subset {H^\infty }

. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if

f
′

f’

is in the Hardy class

H
1

{H^1}

, then

f
∈

M

1

f \in {\mathcal {M}_1}

. Also the theorem is improved to

f
′

∈

H
1

f’ \in {H^1}

implies

f
∈

M

α

f \in {\mathcal {M}_\alpha }

, for all

α
>
0

\alpha > 0

. Finally, let

α
>
0

\alpha > 0

and let

f
f

be holomorphic in

|

z

|

>
1

|z| > 1

. It is known that

f
f

is bounded if and only if its Cesàro sums are uniformly bounded in

|

z

|

≤
1

|z| \leq 1

. This result is generalized using suitable polynomials defined for

α
>
0

\alpha > 0

.
Fourier inequalities with nonradial weights
Authors: Carton-Lebrun C.
Published: June 2012

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper F f left-parenthesis gamma right-parenthesis equals integral Underscript double-struck upper R Superscript n Baseline Endscripts left-parenthesis e Superscript minus 2 i pi gamma bullet x Baseline minus 1 right-parenthesis f left-parenthesis x right-parenthesis d x comma n greater-than 1">

F

f
(
γ
)
=

∫

R

n

(

e

−
2
i
π
γ
∙
x

−
1
)
f
(
x
)

d
x
,
n
>
1

\mathcal {F}\;f(\gamma ) = {\smallint _{{\mathbb {R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x)\,dx,n > 1

, and

u
u

,

v
v

be nonnegative functions. Sufficient conditions are found in order that

‖

F

f
‖

q
,
u

≤
C

‖
f
‖

p
,
v

\left \| \mathcal {F}\;f\right \| _{q,u} \leq C\left \| f\right \| _{p,v}

for all

f
∈

L
v
p

(

R

n

)

f \in L_v^p({\mathbb {R}^n})

. Pointwise and norm approximations of

F

f

\mathcal {F}\;f

are derived. Similar results are obtained when

u
u

is replaced by a measure weight. In the case

v
(
x
)
=

|

x

|

n
(
p
−
1
)

v(x) = |x{|^{n(p - 1)}}

, a counterexample is given which shows that no Fourier inequality can hold for all

f
f

in

L

c
,
0

∞

L_{c,0}^\infty

. Spherical restriction theorems are established. Further conditions for the boundedness of

F

\mathcal {F}

are discussed.
The set of all iterates is nowhere dense in 𝐶([0,1],[0,1])
Authors: Blokh A. M.
Published: June 2012

We prove that if a mixing map

f
:
[
0
,
1
]
→
[
0
,
1
]

f:[0,1] \to [0,1]

belongs to the

C
0

{C^0}

-closure of the set of iterates and

f
(
0
)
≠
0

f(0) \ne 0

,

f
(
1
)
≠
1

f(1) \ne 1

then

f
f

is an iterate itself. Together with some one-dimensional techniques it implies that the set of all iterates is nowhere dense in

C
(
[
0
,
1
]
,
[
0
,
1
]
)

C([0,1],[0,1])

giving the final answer to the question of A. Bruckner, P. Humke and M. Laczkovich.
Actions of linearly reductive groups on PI-algebras
Authors: Vonessen Nikolaus
Published: June 2012

Let

G
G

be a linearly reductive group acting rationally on a

PI

{\text {PI}}

-algebra

R
R

. We study the relationship between

R
R

and the fixed ring

R
G

{R^G}

, generalizing earlier results obtained under the additional hypothesis that

R
R

is affine.
Countable closed 𝐿𝐹𝐶-groups with 𝑝-torsion
Authors: Leinen Felix
Published: June 2012

LetLFCLFCbe the class of all locallyFCFC-groups. We study the existentially closed groups in the classLFCpLF{C_p}of allLFCLFC-groupsHHwhose torsion subgroupT(H)T(H)is app-group. Differently from the situation inLFCLFC, every existentially closedLFCpLF{C_p}-group is already closed inLFCpLF{C_p}, and there exist2ℵ0{2^{{\aleph _0}}}countable closedLFCPLF{C_P}-groupsGG. However, in the countable case,T(G)T(G)is up to isomorphism always a unique locally finitepp-group with similar properties as the unique countable existentially closed locally finitepp-groupEp{E_p}.
Homogeneous foliations of spheres
Authors: Lu Duojia
Published: June 2012

In this paper we discuss Riemannian foliations of the round sphere. We prove that there are no homogeneous Riemannian foliations of the round sphere with dimensions of the leaves bigger than three.
A unique continuation property on the boundary for solutions of elliptic equations
Authors: Jin Zhi Ren
Published: June 2012

We prove the following conclusion: if

u
u

is a harmonic function on a smooth domain

Ω
\Omega

in

R
n

{R^n}

,

n
≥
3

n \geq 3

, or a solution of a general second-order linear elliptic equation on a domain

Ω
\Omega

in

R
2

{R^2}

, and if there are

x
0

∈
∂
Ω

{x_0} \in \partial \Omega

and constants

a
a

,

b
>
0

b > 0

such that

|

u
(
x
)

|

≤
a
exp
⁡
{
−
b

/

|

x
−

x
0

|

}

|u(x)| \leq a\exp \{ - b/|x - {x_0}|\}

for

x
∈
Ω

x \in \Omega

,

|

x
−

x
0

|

|x - {x_0}|

small, then

u
=
0

u = 0

in

Ω
\Omega

. The decay rate in our results is best possible by the example that

u
=

u =

real part of

exp
⁡
{
−
1

/

z
α

}

\exp \{ - 1/{z^\alpha }\}

,

0
>
α
>
1

0 > \alpha > 1

, is harmonic but not identically zero in the right complex half-plane.
Zero-equivalence in function fields defined by algebraic differential equations
Authors: Shackell John
Published: June 2012

We consider function fields obtained as towers over the field of rational functions, each extension being by a solution of an algebraic differential equation. On the assumption that an oracle exists for the constants, we present two algorithms for determining whether a given expression is functionally equivalent to zero in such a field. The first, which uses Gröbner bases, has the advantage of theoretical simplicity, but is liable to involve unnecessary computations. The second method is designed with a view to eliminating these.
The first two obstructions to the freeness of arrangements
Authors: Yuzvinsky Sergey
Published: June 2012

In his previous paper the author characterized free arrangements by the vanishing of cohomology modules of a certain sheaf of graded modules over a polynomial ring. Thus the homogeneous components of these cohomology modules can be viewed as obstructions to the freeness of an arrangement. In this paper the first two obstructions are studied in detail. In particular the component of degree zero of the first nontrivial cohomology module has a close relation to formal arrangements and to the operation of truncation. This enables us to prove that in dimension greater than two every free arrangement is formal and not a proper truncation of an essential arrangement.
Computing the Mordell-Weil rank of Jacobians of curves of genus two
Authors: Gordon Daniel M. , Grant David
Published: June 2012

We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method.
On the theory of Frobenius extensions and its application to Lie superalgebras
Authors: Bell Allen D. , Farnsteiner Rolf
Published: June 2012

By using an approach to the theory of Frobenius extensions that emphasizes notions related to associative forms, we obtain results concerning the trace and corestriction mappings and transitivity. These are employed to show that the extension of enveloping algebras determined by a subalgebra of a Lie superalgebra is a Frobenius extension, and to study certain questions in representation theory.
Critical LIL behavior of the trigonometric system
Authors: Berkes I.
Published: June 2012

It is a classical fact that for rapidly increasing

(

n
k

)

({n_k})

the sequence

(
cos
⁡

n
k

x
)

(\cos {n_k}x)

behaves like a sequence of i.i.d. random variables. Actually, this almost i.i.d. behavior holds if

(

n
k

)

({n_k})

grows faster than

e

c

k

{e^{c\sqrt k }}

; below this speed we have strong dependence. While there is a large literature dealing with the almost i.i.d. case, practically nothing is known on what happens at the critical speed

n
k

∼

e

c

k

{n_k} \sim {e^{c\sqrt k }}

(critical behavior) and what is the probabilistic nature of

(
cos
⁡

n
k

x
)

(\cos {n_k}x)

in the strongly dependent domain. In our paper we study the critical LIL behavior of

(
cos
⁡

n
k

x
)

(\cos {n_k}x)

i.e., we investigate how classical fluctuational theorems like the law of the iterated logarithm and the Kolmogorov-Feller test turn to nonclassical laws in the immediate neighborhood of

n
k

∼

e

c

k

{n_k} \sim {e^{c\sqrt k }}

.
Chains, null-chains, and CR geometry
Authors: Koch Lisa K.
Published: June 2012

A system of distinguished curves distinct from chains is defined on indefinite nondegenerate

CR

{\text {CR}}

hypersurfaces; the new curves are called null-chains. The properties of these curves are explored, and it is shown that two sufficiently nearby points of any nondegenerate

CR

{\text {CR}}

hypersurface can be connected by either a chain or a null-chain.
Parabolic systems: the 𝐺𝐹(3)-case
Authors: Meixner Thomas
Published: June 2012

Parabolic systems defined over

G
F
(
q
)

GF(q)

have been classified by Timmesfeld for

q
≥
4

q \geq 4

and by Stroth for

q
=
2

q = 2

(see references). We deal with the case

q
=
3

q = 3

.
𝐺𝐿(4,𝑅)-Whittaker functions and ₄𝐹₃(1) hypergeometric series
Authors: Stade Eric
Published: June 2012

In this paper we consider spaces of

GL

(
4
,

R

)

{\text {GL}}(4,\mathbb {R})

-Whittaker functions, which are special functions that arise in the study of

GL

(
4
,

R

)

{\text {GL}}(4,\mathbb {R})

automorphic forms. Our main result is to determine explicitly the series expansion for a

GL

(
4
,

R

)

{\text {GL}}(4,\mathbb {R})

-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all

GL

(
4
,

R

)

{\text {GL}}(4,\mathbb {R})

-Whittaker functions of fixed eigenvalues. The series that we find in the case of

GL

(
4
,

R

)

{\text {GL}}(4,\mathbb {R})

is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series— namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of

GL

(
n
,

R

)

{\text {GL}}(n,\mathbb {R})

.
Product formulas and convolutions for angular and radial spheroidal wave functions
Authors: Connett William C. , Markett Clemens , Schwartz Alan L.
Published: June 2012

Product formulas for angular spheroidal wave functions on[0,π][0,\pi ]and for radial spheroidal wave functions on[0,∞)[0,\infty )are presented, which generalize results for the ultraspherical polynomials and functions as well as for the Mathieu functions. Although these functions cannot be given in closed form, the kernels of the product formulas are represented in an explicit, and surprisingly simple way in terms of Bessel functions so that the exact range of positivity can easily be read off. The formulas are used to introduce two families of convolution structures on[0,π][0,\pi ]and[0,∞)[0,\infty ), many of which provide new hypergroups. We proceed from the fact that the spheroidal wave functions are eigenfunctions of Sturm-Liouville equations of confluent Heun type and employ a partial differential equation technique based on Riemann’s integration method.
Algebraic particular integrals, integrability and the problem of the center
Authors: Schlomiuk Dana
Published: June 2012

In this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields. We first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center: these curves control the changes in the systems as parameters vary. The bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered, namely the real four-dimensional projective space. Next, we consider the known four algebraic conditions for the center for quadratic vector fields. One of them says that the system is Hamiltonian, a condition which has a clear geometric meaning. We determine the geometric meaning of the remaining other three algebraic conditions (I), (II), (III). We show that a quadratic system with a weak focus

F
F

, possessing algebraic particular integrals not passing through

F
F

of the following types, satisfies in some coordinate axes the condition (I), (II) or (III) respectively and hence has a center at

F
F

: either a parabola and an irreducible cubic particular integral having only one point at infinity, coinciding with the one of the parabola; or a straight line and an irreducible conic curve; or distinct straight lines (possibly with complex coefficients). We show that each one of these geometric properties is generic for systems satisfying the corresponding algebraic condition for the center. Another version of this result in terms of real algebraic curves is given. These results make clear the many facets of the problem of the center in the quadratic case, in particular the question of integrability and form a basis for analogous investigations for the general problem of the center for cubic systems.
Number of orbits of branch points of 𝑅-trees
Authors: Jiang Renfang
Published: June 2012

An

R
R

-tree is a metric space in which any two points are joined by a unique arc. Every arc is isometric to a closed interval of

R
R

. When a group

G
G

acts on a tree (

Z
Z

-tree)

X
X

without inversion, then

X

/

G

X/G

is a graph. One gets a presentation of

G
G

from the quotient graph

X

/

G

X/G

with vertex and edge stabilizers attached. For a general

R
R

-tree

X
X

, there is no such combinatorial structure on

X

/

G

X/G

. But we can still ask what the maximum number of orbits of branch points of free actions on

R
R

-trees is. We prove the finiteness of the maximum number for a family of groups, which contains free products of free abelian groups and surface groups, and this family is closed under taking free products with amalgamation.
The cohomology algebra of a commutative group scheme
Authors: Fossum Robert , Haboush William
Published: June 2012

Let

k
k

be a commutative ring with unit of characteristic

p
>
0

p > 0

and let

G
=
Spec
⁡
(
A
)

G = \operatorname {Spec}(A)

be an affine commutative group scheme over

k
k

. Let

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

be the graded Hochschild algebraic group cohomology algebra and, for

M
M

a rational

G
G

-module, let

H

∙

(
G
,
M
)

{{\text {H}}^ \bullet }(G,M)

denote the graded Hochschild cohomology

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

-module. We show that

H

∙

(
G
)

{{\text {H}}^ \bullet }(G)

is, in general, a graded Hopf algebra. When

G
=

G

a
,
k

G = {{\mathbf {G}}_{a,k}}

, let

α

p
ν

{\alpha _{{p^\nu }}}

denote the subgroup of

p
ν

{p^\nu }

-nilpotents and let

F

ν

{{\text {F}}_\nu }

denote the

ν
\nu

th power of the Frobenius. We show that for any finite

M
M

that there is a

ν
\nu

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript bullet Baseline left-parenthesis bold upper G Subscript a comma k Baseline comma upper M right-parenthesis approximately-equals upper H Superscript bullet Baseline left-parenthesis alpha Subscript p Sub Superscript nu Subscript Baseline comma upper M right-parenthesis circled-times Subscript k Baseline upper F Subscript nu Superscript asterisk Baseline left-parenthesis upper H Superscript bullet Baseline left-parenthesis bold upper G Subscript a comma k Baseline right-parenthesis right-parenthesis">

H

∙

(

G

a
,
k

,
M
)
≅

H

∙

(

α

p
ν

,
M
)

⊗
k

F

ν

∗

(

H

∙

(

G

a
,
k

)
)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M) \cong {{\text {H}}^ \bullet }({\alpha _{{p^\nu }}},M){ \otimes _k}{\text {F}}_{\nu }^\ast ({{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}}))

\]
where

F

ν
∗

{\text {F}}_\nu ^\ast

is the endomorphism of

H

∙

(

G

a
,
k

)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})

induced by

F
v

{F_v}

. As a consequence, we can show that

H

∙

(

G

a
,
k

,
M
)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M)

is a finitely generated module over

H

∙

(

G

a
,
k

)

{{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})

when

M
M

is a finite dimensional vector space over

k
k

.
On the evaluation map
Authors: Murillo Aniceto
Published: June 2012

The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geometric implications: (i) A

1
1

-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let

E

→

ρ

B

E\xrightarrow {\rho }B

be a fibration of simplyconnected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if

ρ
\rho

is surjective in rational homotopy and the evaluation map of

E
E

is nontrivial, then the evaluation map of the fibre is different from zero.
Geometric curvature bounds in Riemannian manifolds with boundary
Authors: Alexander Stephanie B. , Berg I. David , Bishop Richard L.
Published: June 2012

An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yau’s isoperimetric inequality for spaces with negative curvature are generalized.
Intersection cohomology of 𝑆¹-actions
Authors: Hector Gilbert , Saralegi Martin
Published: June 2012

Given a free action of the circle

S

1

{{\mathbf {S}}^1}

on a differentiable manifold

M
M

, there exists a long exact sequence that relates the cohomology of

M
M

with the cohomology of the manifold

M

/

S

1

M/{{\mathbf {S}}^1}

. This is the Gysin sequence. This result is still valid if we allow the action to have stationary points. In this paper we are concerned with actions where fixed points are allowed. Here the quotient space

M

/

S

1

M/{{\mathbf {S}}^1}

is no longer a manifold but a stratified pseudomanifold (in terms of Goresky and MacPherson). We get a similar Gysin sequence where the cohomology of

M

/

S

1

M/{{\mathbf {S}}^1}

is replaced by its intersection cohomology. As in the free case, the connecting homomorphism is given by the product with the Euler class

[
e
]

[e]

. Also, the vanishing of this class is related to the triviality of the action. In this Gysin sequence we observe the phenomenon of perversity shifting. This is due to the allowability degree of the Euler form.
The finite part of singular integrals in several complex variables
Authors: Wang Xiaoqin
Published: June 2012

A divergent integral can sometimes be handled by assigning to it as its value the finite part in the sense of Hadamard. This is done by expanding the integral over the complement of a symmetric neighborhood of a singularity in powers of the radius, and throwing away the negative powers. In this paper the finite part of a singular integral of Cauchy type is defined, and this is then used to describe the boundary behavior of derivatives of a Cauchy-type integral. The finite part of a singular integral of Bochner-Martinelli type is studied, and an extension of the Plemelj jump formulas is shown to hold.
Hypersurfaces with constant mean curvature in the complex hyperbolic space
Authors: Fornari Susana , Frensel Katia , Ripoll Jaime
Published: June 2012

A classical theorem of A. D. Alexandrov characterized round spheres is extended to the complex hyperbolic space

C

H

2

{\mathbf {C}}{{\mathbf {H}}^2}

of constant holomorphic sectional curvature. A detailed description of the horospheres and equidistant hypersurfaces in

C

H

2

{\mathbf {C}}{{\mathbf {H}}^2}

determining in particular their stability, is also given.
Zeros of the successive derivatives of Hadamard gap series
Authors: Gethner Robert M.
Published: June 2012

A complex number

z
z

is in the final set of an analytic function

f
f

, as defined by Pólya, if every neighborhood of

z
z

contains zeros of infinitely many

f

(
n
)

{f^{(n)}}

. If

f
f

is a Hadamard gap series, then the part of the final set in the open disk of convergence is the origin along with a union of concentric circles.
Complex geodesics and iterates of holomorphic maps on convex domains in 𝐶ⁿ
Authors: Mercer Peter R.
Published: June 2012

We study complex geodesics

f
:
Δ
→
Ω

f:\Delta \to \Omega

, where

Δ
\Delta

is the unit disk in

C

{\mathbf {C}}

and

Ω
\Omega

belongs to a class of bounded convex domains in

C

n

{{\mathbf {C}}^n}

with no boundary regularity assumption. Along with continuity up to the boundary, existence of such complex geodesics with two prescribed values

z
z

,

w
∈

Ω
¯

w \in \bar \Omega

is established. As a consequence we obtain some new results from iteration theory of holomorphic self maps of bounded convex domains in

C

n

{{\mathbf {C}}^n}

.
A metric deformation and the first eigenvalue of Laplacian on 1-forms
Authors: Otofuji Takashi
Published: June 2012

We search for a higher-dimensional analogue of Calabi’s example of a metric deformation, quoted by Cheeger, which inspired him to prove an inequality between the first eigenvalue of the Laplacian on functions and an isoperimetric constant. We construct an example of a metric deformation on

S
n

{S^n}

,

n

≥
5

{n} \geq 5

, where the first eigenvalue of the Laplacian on functions remains bounded above from zero, and the first eigenvalue of the Laplacian on

1
1

-forms tends to zero. This metric deformation makes the sphere in the limit into a manifold with a cone singularity, which is an intermediate point on a path of deformation from an (

S
n

{S^n}

, some metric) to an (

S

n
−
1

×

S
1

{S^{n - 1}} \times {S^1}

, some metric).
The constrained least gradient problem in 𝑅ⁿ
Authors: Sternberg Peter , Williams Graham , Ziemer William P.
Published: June 2012

We consider the constrained least gradient problem
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="inf left-brace integral Underscript normal upper Omega Endscripts StartAbsoluteValue nabla u EndAbsoluteValue d x colon u element-of upper C Superscript 0 comma 1 Baseline left-parenthesis normal upper Omega overbar right-parenthesis comma StartAbsoluteValue nabla u EndAbsoluteValue less-than-or-equal-to 1 a period e period comma u equals g on partial-differential normal upper Omega right-brace">

inf

{

∫
Ω

|

∇
u

|

d
x
:
u
∈

C

0
,
1

(

Ω
¯

)
,

|

∇
u

|

≤
1

a.e.

,
u
=
g

on

∂
Ω

}

\inf \left \{ {\int _\Omega {|\nabla u|dx:u \in {C^{0,1}}(\bar \Omega ),\quad |\nabla u| \leq 1\;{\text {a.e.}},u = g\;{\text {on}}\;\partial \Omega } } \right \}

\]
which arises as the relaxation of a nonconvex problem in optimal design. We establish the existence of a solution by an explicit construction in which each level set is required to solve an obstacle problem. We also establish the uniqueness of solutions and discuss their structure.
Lyapunov graphs and flows on surfaces
Authors: de Rezende K. A. , Franzosa R. D.
Published: June 2012

In this paper, a characterization of Lyapunov graphs associated to smooth flows on surfaces is presented. We first obtain necessary and sufficient conditions for a Lyapunov graph to be associated to Morse-Smale flows and then generalize them to smooth flows. The methods employed in the proofs are of interest in their own right for they introduce the use of the Conley index in this context. Moreover, an algorithmic geometric construction of flows on surfaces is described.
Extending the 𝑡-design concept
Authors: Calderbank A. R. , Delsarte P.
Published: June 2012

Let

B

\mathfrak {B}

be a family of

k
k

-subsets of a

v
v

-set

V
V

, with

1
≤
k
≤
v

/

2

1 \leq k \leq v/2

. Given only the inner distribution of

B

\mathfrak {B}

, i.e., the number of pairs of blocks that meet in

j
j

points (with

j
=
0
,
1
,
…
,
k

j = 0,1, \ldots ,k

), we are able to completely describe the regularity with which

B

\mathfrak {B}

meets an arbitrary

t
t

-subset of

V
V

, for each order

t
t

(with

1
≤
t
≤
v

/

2

1 \leq t \leq v/2

). This description makes use of a linear transform based on a system of dual Hahn polynomials with parameters

v
v

,

k
k

,

t
t

. The main regularity parameter is the dimension of a well-defined subspace of

R

t
+
1

{\mathbb {R}^{t + 1}}

, called the

t
t

-form space of

B

\mathfrak {B}

. (This subspace coincides with

R

t
+
1

{\mathbb {R}^{t + 1}}

if and only if

B

\mathfrak {B}

is a

t
t

-design.) We show that the

t
t

-form space has the structure of an ideal, and we explain how to compute its canonical generator.
Completions and fibrations for topological monoids
Authors: Lima-Filho Paulo
Published: June 2012

We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion

M
+

{M^ + }

of a monoid

M
M

in that class and the topologized Grothendieck group

M
~

\tilde M

associated to

M
M

. The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite

CW

{\text {CW}}

-complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naïve approach to Lawson homology [Fri91, LF91a]. Another proof of the Eilenberg-Steenrod axioms for the functors

X
↦

S
P

~

(
X
)

X \mapsto {\tilde {SP}}(X)

in the category of finite

CW

{\text {CW}}

-complexes (Dold-Thom theorem [DT56]) is obtained.
Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups
Authors: Kono Akira , Yagita Nobuaki
Published: June 2012

The Steenrod algebra structures of

H
∗

(
B
G
;
Z

/

p
)

{H^\ast }(BG;Z/p)

for compact Lie groups are studied. Using these, Brown-Peterson cohomology and Morava

K
K

-theory are computed for many concrete cases. All these cases have properties similar as torsion free Lie groups or finite groups, e.g.,

B

P

o
d
d

(
B
G
)
=
0

B{P^{odd}}(BG) = 0

.
Asymptotic behavior for a coalescence problem
Authors: Bruno Oscar , Friedman Avner , Reitich Fernando
Published: June 2012

Consider spherical particles of volume

x
x

having paint on a fraction

y
y

of their surface area. The particles are assumed to be homogeneously distributed at each time

t
t

, so that one can introduce the density number

n
(
x
,
y
,
t
)

n(x,y,t)

. When collision between two particles occurs, the particles will coalesce if and only if they happen to touch each other, at impact, at points which do not belong to the painted portions of their surfaces. Introducing a dynamics for this model, we study the evolution of

n
(
x
,
y
,
t
)

n(x,y,t)

and, in particular, the asymptotic behavior of the mass

x
n
(
x
,
y
,
t
)
d
x

xn(x,y,t)dx

as

t
→
∞

t \to \infty

.
Periodic seeded arrays and automorphisms of the shift
Authors: Brown Ezra
Published: June 2012

The automorphism group

Aut
⁡
(

Σ
2

)

\operatorname {Aut}({\Sigma _2})

of the full

2
2

-shift is conjectured to be generated by the shift and involutions. We approach this problem by studying a certain family of automorphisms whose order was unknown, but which we show to be finite and for which we find factorizations as products of involutions. The result of this investigation is the explicit construction of a subgroup

H

\mathcal {H}

of

Aut
⁡
(

Σ
2

)

\operatorname {Aut}({\Sigma _2})

;

H

\mathcal {H}

is generated by certain involutions

g
n

{g_n}

, and turns out to have a number of curious properties. For example,

g
n

{g_n}

and

g
k

{g_k}

commute unless

n
n

and

k
k

are consecutive integers, the order of

g

n
+
k

∘
⋯
∘

g
k

{g_{n + k}} \circ \cdots \circ {g_k}

is independent of

k
k

, and

H

\mathcal {H}

contains elements of all orders. The investigation is aided by the development of results about certain new types of arrays of

0
0

’s and

1
1

’s called periodic seeded arrays, as well as the use of Boyle and Krieger’s work on return numbers and periodic points.
Threshold growth dynamics
Authors: Gravner Janko , Griffeath David
Published: June 2012

We study the asymptotic shape of the occupied region for monotone deterministic dynamics in

d
d

-dimensional Euclidean space parametrized by a threshold

θ
>
0

\theta > 0

, and a Borel set

N

⊂

R

d

\mathcal {N} \subset {\mathbb {R}^d}

with positive and finite Lebesgue measure. If

A
n

{A_n}

denotes the oocupied set of the dynamics at integer time

n
n

, then

A

n
+
1

{A_{n + 1}}

is obtained by adjoining any point

x
x

for which the volume of overlap between

x
+

N

x + \mathcal {N}

and

A
n

{A_n}

exceeds

θ
\theta

. Except in some degenerate cases, we prove that

n

−
1

A
n

{n^{ - 1}}{A_n}

converges to a unique limiting "shape"

L
L

starting from any bounded initial region

A
0

{A_0}

that is suitably large. Moreover,

L
L

is computed as the polar transform for

1

/

w

1/w

, where

w
w

is an explicit width function that depends on

N

\mathcal {N}

and

θ
\theta

. It is further shown that

L
L

describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of box

(

l
∞

)

({l^\infty })

neighborhoods on

Z

2

{\mathbb {Z}^2}

, these limiting shapes are calculated and the dependence of their anisotropy on

θ
\theta

is examined. Other specific two- and three-dimensional examples are also discussed in some detail.
Rational approximations to the dilogarithm
Authors: Hata Masayoshi
Published: June 2012

The irrationality proof of the values of the dilogarithmic function

L
2

(
z
)

{L_2}(z)

at rational points

z
=
1

/

k

z = 1/k

for every integer

k
∈
(
−
∞
,
−
5
]
∪
[
7
,
∞
)

k \in ( - \infty , - 5] \cup [7,\infty )

is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of

L
2

(
1

/

k
)

{L_2}(1/k)

. Moreover, the linear independence over

Q

{\mathbf {Q}}

of the numbers

1
1

,

log
⁡
(
1
−
1

/

k
)

\log (1 - 1/k)

, and

L
2

(
1

/

k
)

{L_2}(1/k)

is also obtained for each integer

k
∈
(
−
∞
,
−
5
]
∪
[
7
,
∞
)

k \in ( - \infty , - 5] \cup [7,\infty )

.
Universal cover of Salvetti’s complex and topology of simplicial arrangements of hyperplanes
Authors: Paris Luis
Published: June 2012

Let

V
V

be a real vector space. An arrangement of hyperplanes in

V
V

is a finite set

A

\mathcal {A}

of hyperplanes through the origin. A chamber of

A

\mathcal {A}

is a connected component of

V
−
(

∪

H
∈

A

H
)

V - ({ \cup _{H \in \mathcal {A}}}H)

. The arrangement

A

\mathcal {A}

is called simplicial if

∩

H
∈

A

H
=
{
0
}

{ \cap _{H \in \mathcal {A}}}H = \{ 0\}

and every chamber of

A

\mathcal {A}

is a simplicial cone. For an arrangement

A

\mathcal {A}

of hyperplanes in

V
V

, we set
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M left-parenthesis script upper A right-parenthesis equals upper V Subscript double-struck upper C Baseline minus left-parenthesis union Underscript upper H element-of script upper A Endscripts upper H Subscript double-struck upper C Baseline right-parenthesis comma">

M
(

A

)
=

V

C

−

(

⋃

H
∈

A

H

C

)

,

M(\mathcal {A}) = {V_\mathbb {C}} - \left ({\bigcup \limits _{H \in \mathcal {A}} {{H_\mathbb {C}}} } \right ),

\]
where

V

C

=

C

⊗
V

{V_\mathbb {C}} = \mathbb {C} \otimes V

is the complexification of

V
V

, and, for

H
∈

A

H \in \mathcal {A}

,

H

C

{H_\mathbb {C}}

is the complex hyperplane of

V

C

{V_\mathbb {C}}

spanned by

H
H

. Let

A

\mathcal {A}

be an arrangement of hyperplanes of

V
V

. Salvetti constructed a simplicial complex

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

and proved that

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

has the same homotopy type as

M
(

A

)

M(\mathcal {A})

. In this paper we give a new short proof of this fact. Afterwards, we define a new simplicial complex

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

and prove that there is a natural map

p
:

Sal
^

(

A

)
→
Sal
⁡
(

A

)

p:\hat {\operatorname {Sal}}(\mathcal {A}) \to \operatorname {Sal}(\mathcal {A})

which is the universal cover of

Sal
⁡
(

A

)

\operatorname {Sal}(\mathcal {A})

. At the end, we use

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

to give a new proof of Deligne’s result: "if

A

\mathcal {A}

is a simplicial arrangement of hyperplanes, then

M
(

A

)

M(\mathcal {A})

is a

K
(
π
,
1
)

K(\pi ,1)

space." Namely, we prove that

Sal
^

(

A

)

\hat {\operatorname {Sal}}(\mathcal {A})

is contractible if

A

\mathcal {A}

is a simplicial arrangement.
Finitely decidable congruence modular varieties
Authors: Jeong Joohee
Published: June 2012

A class

V

\mathcal {V}

of algebras of the same type is said to be finitely decidable iff the first order theory of the class of finite members of

V

\mathcal {V}

is decidable. Let

V

\mathcal {V}

be a congruence modular variety. In this paper we prove that if

V

\mathcal {V}

is finitely decidable, then the following hold. (1) Each finitely generated subvariety of

V

\mathcal {V}

has a finite bound on the cardinality of its subdirectly irreducible members. (2) Solvable congruences in any locally finite member of

V

\mathcal {V}

are abelian. In addition we obtain various necessary conditions on the congruence lattices of finite subdirectly irreducible algebras in

V

\mathcal {V}

.
The semigroup property of value functions in Lagrange problems
Authors: Wolenski Peter R.
Published: June 2012

The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at

t
=
0

t = 0

. The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When

L
L

is in addition finite-valued, the results are given in terms of pointwise convergence.
On a conjecture regarding nonstandard uniserial modules
Authors: Eklof Paul C. , Shelah Saharon
Published: June 2012

We consider the question of which valuation domains (of cardinality

ℵ
1

{\aleph _1}

) have nonstandard uniserial modules. We show that a criterion conjectured by Osofsky is independent of

ZFC

+

GCH

{\text {ZFC}} + {\text {GCH}}

.
Normality in 𝑋² for compact 𝑋
Authors: Gruenhage G. , Nyikos P. J.
Published: June 2012

In 1977, the second author announced the following consistent negative answer to a question of Katětov: Assuming

MA

+
¬

CH

{\text {MA}} + \neg {\text {CH}}

, there is a compact nonmetric space

X
X

such that

X
2

{X^2}

is hereditarily normal. We give the details of this example, and construct another example assuming

CH

{\text {CH}}

. We show that both examples can be constructed so that

X
2

∖
Δ

{X^2}\backslash \Delta

is perfectly normal. We also construct in

ZFC

{\text {ZFC}}

a compact nonperfectly normal

X
X

such that

X
2

∖
Δ

{X^2}\backslash \Delta

is normal.