bluuebunny/crossref_metadata_2025 huggingface dataset:
Weakening the topology of a Lie group
Authors: Stevens T. Christine
Published: June 2010

With any topological group

(
G
,

U

)

(G, \mathcal {U})

one can associate a locally arcwise-connected group

(
G
,

U

∗

)

(G, {\mathcal {U}}^{\ast })

, where

U

∗

{\mathcal {U}}^{\ast }

is stronger than

U

\mathcal {U}

.

(
G
,

U

)

(G, \mathcal {U})

is a weakened Lie

(
W
L
)

(WL)

group if

(
G
,

U

∗

)

(G, {\mathcal {U}}^{\ast })

is a Lie group. In this paper the author shows that the WL groups with which a given connected Lie group

(
L
,

J

)

(L,\mathcal {J})

is associated are completely determined by a certain abelian subgroup

H
H

of

L
L

which is called decisive. If

L
L

has closed adjoint image, then

H
H

is the center

Z
(
L
)

Z(L)

of

L
L

; otherwise,

H
H

is the product of a vector group

V
V

and a group

J
J

that contains

Z
(
L
)

Z(L)

.

J

/

Z
(
L
)

J/Z(L)

is finite (trivial if

L
L

is solvable). We also discuss the connection between these theorems and recent results of Goto.
Interpolating sequences for 𝑄𝐴_{𝐵}
Authors: Sundberg Carl , Wolff Thomas H.
Published: June 2010

Let

B
B

be a closed algebra lying between

H
∞

{H^\infty }

and

L
∞

{L^\infty }

of the unit circle. We define

Q

A
B

=

H
∞

∩

B
¯

QA_B = H^\infty \cap \bar {B}

, the analytic functions in

Q
B

=
B
∩

B
¯

Q_B = B \cap \bar {B}

. By work of Chang,

Q
B

{Q_B}

is characterized by a vanishing mean oscillation condition. We characterize the sequences of points

{

z
n

}

\left \{{{z_n}} \right \}

in the open unit disc for which the interpolation problem

f
(

z
n

)
=

λ
n

,
n
=
1
,
2
,
…

f({z_n}) = {\lambda _n}, n = 1, 2,\ldots

, is solvable with

f
∈

Q
B

f \in {Q_B}

for any bounded sequence of numbers

{

λ
n

}

\left \{{{\lambda _n}} \right \}

. Included as a necessary part of our proof is a study of the algebras

Q

A
B

Q{A_B}

and

Q
B

{Q_B}

.
On genus 2 Heegaard diagrams for the 3-sphere
Authors: Kaneto Takeshi
Published: June 2010

Let

D
D

be any genus

2
2

Heegaard diagram for the

3
3

-sphere and

⟨

a
1

,

a
2

;

r
~

1

,

r
~

2

⟩

\left \langle {{a_1}, {a_2}; {{\tilde r}_1}, {{\tilde r}_2}} \right \rangle

be the cyclically reduced presentation associated with

D
D

. We shall show that

r
~

1

{{\tilde {r}}_1}

contains

r
~

2

{{\tilde {r}}_2}

or

r
~

2

−
1

{\tilde {r}}_2^{-1}

as a subword in cyclic sense if

{

r
~

1

,

r
~

2

}

≠

{

a
1

±
1

,

a
2

±
1

}

\left \{{\tilde r}_1, {\tilde r}_2 \right \} \ne \left \{{a_1}^{\pm 1}, {a_2}^{\pm 1} \right \}

holds, and that, using this property,

⟨

a
1

,

a
2

;

r
1

,

r
2

⟩

\left \langle {a_1}, {a_2};{r_1}, {r_2} \right \rangle

can be transformed to the trivial one

⟨

a
1

,

a
2

;

a
1

±
1

,

a
2

±
1

⟩

\left \langle {{a_1}, {a_2};{a_1}^{\pm 1}, a_2^{\pm 1}} \right \rangle

. By the recent positive solution of genus

2
2

Poincaré conjecture, our result implies the purely algebraic, algorithmic solution to the decision problem; whether a given

3
3

-manifold with a genus

2
2

Heegaard splitting is simply connected or not, equivalently, is homeomorphic to the

3
3

-sphere or not.
Bundle-like foliations with Kählerian leaves
Authors: Escobales Richard H.
Published: June 2010

For bundle-like foliations with Kählerian leaves a certain function

f
f

is studied and its Laplacian along a leaf is computed. From this computation one obtains geometric conditions which guarantee the integrability of the distribution orthogonal to that determined by the leaves. When the leaves are compact, the key condition needed to guarantee the integrability of this orthogonal distribution can be interpreted as a condition on the first Chern class of each of the leaves.
Completion of Akahori’s construction of the versal family of strongly pseudoconvex CR structures
Authors: Miyajima Kimio
Published: June 2010

Let

M
M

be a compact smooth boundary of a strongly pseudo-convex domain of a complex manifold

N
N

with dim

N
⩾
4

N \geqslant 4

. We established a sharp a priori estimate for the Laplacian operator associated with Akahori’s subcomplex of the

T
′

N

|

M

T’{N_{|M}}

-valued

∂
¯

b

{\bar \partial _b}

-complex to construct the complex analytic versal family (in the sense of Kuranishi) of

C
R

CR

structures of class

C
∞

{C^\infty }

on

M
M

.
Rees matrix covers for locally inverse semigroups
Authors: McAlister D. B.
Published: June 2010

A regular semigroup

S
S

is locally inverse if each local submonoid

e
S
e

eSe

,

e
e

an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism

θ
\theta

which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.
Cocycles and local product decomposition
Authors: Tanaka Jun-ichi
Published: June 2010

As an application of cocycles, we establish a relation between the classical Hardy spaces on the real line

R
R

and simply invariant subspaces on a quotient of the Bohr group. When this result is specialized suitably, it yields the well-known results concerning the elements of invariant subspaces. We also study, by using Gamelin’s representation theorem, unitary functions which are the values of cocycles.
Separatrix and limit cycles of quadratic systems and Dulac’s theorem
Authors: Chicone Carmen , Shafer Douglas S.
Published: June 2010

Separatrix cycles for a planar quadratic vector field are studied. The results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles.
Subcontinua with degenerate tranches in hereditarily decomposable continua
Authors: Oversteegen Lex G. , Tymchatyn E. D.
Published: June 2010

A hereditarily decomposable, irreducible, metric continuum

M
M

admits a mapping

f
f

onto

[
0
,
1
]

[0,1]

such that each

f

−
1

(
t
)

{f^{ - 1}}(t)

is a nowhere dense subcontinuum. The sets

f

−
1

(
t
)

{f^{ - 1}}(t)

are the tranches of

M
M

and

f

−
1

(
t
)

{f^{ - 1}}(t)

is a tranche of cohesion if

t
∈
{
0
,
1
}

t \in \{ 0,1\}

or
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript negative 1 Baseline left-parenthesis t right-parenthesis equals upper C 1 left-parenthesis f Superscript negative 1 Baseline left-parenthesis left-bracket 0 comma t right-parenthesis right-parenthesis right-parenthesis intersection upper C 1 left-parenthesis f Superscript negative 1 Baseline left-parenthesis left-parenthesis t comma 1 right-bracket right-parenthesis right-parenthesis">

f

−
1

(
t
)
=

C1

(

f

−
1

(
[
0
,
t
)
)
)
∩

C1

(

f

−
1

(
(
t
,
1
]
)
)

{f^{ - 1}}(t) = {\text {C1}}({f^{ - 1}}([0,t))) \cap {\text {C1}}\,({f^{ - 1}}((t,1]))

. The following answer a question of Mahavier and of E. S. Thomas, Jr. Theorem. Every hereditarily decomposable continuum contains a subcontinuum with a degenerate tranche. Corollary. If in an irreducible hereditarily decomposable continuum each tranche is nondegenerate then some tranche is not a tranche of cohesion. The theorem answers a question of Nadler concerning arcwise accessibility in hyperspaces.
Fully nonlinear, uniformly elliptic equations under natural structure conditions
Authors: Trudinger Neil S.
Published: June 2010

We derive first and second derivative estimates for classical solutions of fully nonlinear, uniformly elliptic equations which are subject to natural structure conditions analogous to those proposed and treated by Ladyzhenskaya and Ural’tseva for quasilinear equations. As an application we extend recent work of Evans and Lions on the Bellman equation for families of linear operators to families of quasilinear operators.
On the location of zeros of oscillatory solution
Authors: Gingold H.
Published: June 2010

The location of zeros of solutions of second order singular differential equations is provided by a new asymptotic decomposition formula. The approximate location of zeros is provided with high accuracy error estimates in the neighbourhood of the point at infinity. The same asymptotic formula suggested is applicable to the neighbourhood of most types of singularities as well as to the neighbourhoods of regular points.
Time-ordered operators. II
Authors: Gill Tepper L.
Published: June 2010

In this paper, we substantially improve on the work of [G1]. After constructing the general mathematical foundations for linear time-ordered evolution equations, we apply our results to show that both the perturbation expansion and the Feynman diagram method are mathematically sound. We provide a remainder term so that the expansion may be considered exact at all orders. We then show that time-ordered operators naturally induce an operator-valued path integral whenever a transition kernel is given.
Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders
Authors: Fabes Eugene , Salsa Sandro
Published: June 2010

In this paper the authors prove unique solvability of the initial-Dirichlet problem for the heat equation in a cylindrical domain with Lipschitz base, lateral data in

L
p

,
p
⩾
2

{L^p},p \geqslant 2

, and zero initial values. A Poisson kernel for this problem is shown to exist with the property that its

L
2

{L^2}

-averages over parabolic rectangles are equivalent to

L
1

{L^1}

-averages over the same sets.
Preorderings compatible with probability measures
Authors: Chuaqui Rolando , Malitz Jerome
Published: June 2010

The main theorem proved in this paper is: Let

B
B

be a

σ
\sigma

-complete Boolean algebra and

≽
a

\succcurlyeq a

binary relation with field

B
B

such that: (i) Every finite subalgebra

B
′

B’

admits a probability measure

μ
′

\mu ’

such that for

p
,
q
∈

B
′

,
p
≽
q

i
f
f

μ
′

p
⩾

μ
′

q

p,q \in B’,p \succcurlyeq q\;iff\mu ’p \geqslant \mu ’q

. (ii) If for every

i
,

p
i

,
q
∈
B

i,{p_i},q \in B

and

p
i

⊆

p

i
+
1

≼
q

{p_i} \subseteq {p_{i + 1}} \preccurlyeq q

, then

∪

i
>
∞

p
i

≼
q

{ \cup _{i > \infty }}{p_i} \preccurlyeq q

. Under these conditions there is a

σ
\sigma

-additive probability measure

μ
\mu

on

B
B

such that: (a) If there is

a

p
∈
B

a\;p \in B

, such that for every

q
⊆
p

q \subseteq p

there is a

q
′

⊆
q

q’ \subseteq q

with

q
′

≼
q
,

q
′

⋠
0

q’ \preccurlyeq q,q’ \npreceq 0

, and

q
⋠

q
′

q \npreceq q’

, then we have that for every

p
,
q
∈
B
,
μ

p
⩾
μ

q

i
f
f

p
≽
q

p,q \in B,\mu \,p \geqslant \mu \,q\,iff\,p \succcurlyeq q

. (b) If for every

p
∈
B

p \in B

, there is

a

q
⊆
p

a\;q \subseteq p

such that

q
′

⊆
q

q’ \subseteq q

implies

q
≼

q
′

o
r

q
′

≼
0

q \preccurlyeq q’\;or\;q’ \preccurlyeq 0

, then we have that for every

p
,
q
∈
B
,
p
≽
q

p,q \in B,p \succcurlyeq q

implies

μ
p
⩾
μ
q

\mu p \geqslant \mu q

.
On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-Radon partitions, and orientations of graphs
Authors: Greene Curtis , Zaslavsky Thomas
Published: June 2010

The doubly indexed Whitney numbers of a finite, ranked partially ordered set

L
L

are (the first kind)

w

i
j

=
∑

{
μ
(

x
i

,

x
j

)
:

x
i

,

x
j

∈
L

{w_{ij}} = \sum {\{ \mu ({x^i},{x^j}):{x^i},{x^j} \in L}

with ranks

i
,
j
}

i,j\}

and (the second kind)

W

i
j

=

{W_{ij}} =

the number of

(

x
i

,

x
j

)

({x^i},{x^j})

with

x
i

⩽

x
j

{x^i} \leqslant {x^j}

. When

L
L

has a

0
0

element, the ordinary (simply indexed) Whitney numbers are

w
j

=

w

0
j

{w_j} = {w_{0j}}

and

W
j

=

W

0
j

=

W

j
j

{W_j} = {W_{0j}} = {W_{jj}}

. Building on work of Stanley and Zaslavsky we show how to interpret the magnitudes of Whitney numbers of geometric lattices and semilattices arising in geometry and graph theory. For example: The number of regions, or of

k
k

-dimensional faces for any

k
k

, of an arrangement of hyperplanes in real projective or affine space, that do not meet an arbitrary hyperplane in general position. The number of vertices of a zonotope

P
P

inside the visible boundary as seen from a distant point on a generating line of

P
P

. The number of non-Radon partitions of a Euclidean point set not due to a separating hyperplane through a fixed point. The number of acyclic orientations of a graph (Stanley’s theorem, with a new, geometrical proof); the number with a fixed unique source; the number whose set of increasing arcs (in a fixed ordering of the vertices) has exactly

q
q

sources (generalizing Rényi’s enumeration of permutations with

q
q

"outstanding" elements). The number of totally cyclic orientations of a plane graph in which there is no clockwise directed cycle. The number of acyclic orientations of a signed graph satisfying conditions analogous to an unsigned graph’s having a unique source.
Some prime elements in the lattice of interpretability types
Authors: Pudlák Pavel
Published: June 2010

A general theorem is proved which implies that the types of

PA
\text {PA}

(Peano Arithmetic),

ZF
\text {ZF}

(Zermelo-Fraenkel Set Theory) and

GB
\text {GB}

(Gödel-Bernays Set Theory) are prime in the lattice of interpretability types.
Degenerate elliptic operators as regularizers
Authors: Pederson R. N.
Published: June 2010

The spaces

K

m
k

{\mathcal {K}_{mk}}

, introduced in the Nehari Volume of Journal d’Analyse Mathématique, for nonnegative integer values of

m
m

and arbitrary real values of

k
k

are extended to negative values of

m
m

. The extension is consistent with the equivalence

∥

ζ
j

u

∥

m
,
k

∼∥
u

∥

m
,
k
−
j

\parallel {\zeta ^j}u{\parallel _{m,k}}\sim \parallel u{\parallel _{m,k - j}}

, the inequality

∥

D
α

u

∥

m
,
k

⩽

const

∥
u

∥

m
+

|

α

|

,
k
+

|

α

|

\parallel {D^\alpha }u{\parallel _{m,k}} \leqslant {\text {const}}\parallel u{\parallel _{m + |\alpha |,k + |\alpha |}}

, and the generalized Cauchy-Schwarz inequality

|

⟨

u
,
v

⟩

|

⩽∥
u

∥

m
,
k

∥
v

∥

−
m
,
−
k

|\langle {u,v} \rangle | \leqslant \parallel u\,{\parallel _{m,k}}\parallel v\parallel _{ - m, - k}

. (Here

⟨
u
,
υ
⟩

\langle u, \upsilon \rangle

is the

L
2

{L_2}

scalar product.) There exists a second order degenerate elliptic operator which maps

K

m
,
k

1
−
1

{\mathcal {K}_{m,k}}\,1 - 1

onto

K

m
−
2
,
k

{\mathcal {K}_{m - 2,k}}

. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)
All varieties of central completely simple semigroups
Authors: Petrich Mario , Reilly Norman R.
Published: June 2010

Completely simple semigroups may be considered as a variety of algebras with the binary operation of multiplication and the unary operation of inversion. A completely simple semigroup is central if the product of any two idempotents lies in the centre of the containing maximal subgroup. Central completely simple semigroups form a subvariety

C

\mathcal {C}

of the variety of all completely simple semigroups. We find an isomorphic copy of

L

(

C

)

\mathcal {L}(\mathcal {C})

as a subdirect product of the lattices

L

(

R

B

)

\mathcal {L}(\mathcal {R}\,\mathcal {B})

,

L

(

A

G

)

\mathcal {L}(\mathcal {A}\,\mathcal {G})

, and

L

(

G

)

\mathcal {L}(\mathcal {G})

of all varieties of rectangular bands, abelian groups, and groups, respectively. We consider also several homomorphisms and study congruences they induce.
Forcing positive partition relations
Authors: Todorčević Stevo
Published: June 2010

We show how to force two strong positive partition relations on

ω
1

{\omega _1}

and use them in considering several well-known open problems.
Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator
Authors: Sawyer Eric
Published: June 2010

Characterizations are obtained for those pairs of weight functions

w
,
υ

w,\upsilon

for which the Hardy operator

T
f
(
x
)
=

∫
0
x

f
(
s
)

d
s

Tf(x) = \int _0^x {f(s)\;ds}

is bounded from the Lorentz space

L

r
,
s

(
(
0
,
∞
)
,
υ

d
x
)

{L^{r,s}}((0,\infty ),\upsilon \,dx)

to

L

p
,
q

(
(
0
,
∞
)
,
w

d
x
)
,
0
>
p
,
q
,
r
,
s
⩽
∞

{L^{p,q}}((0,\infty ),w\,dx),0 > p,q,r,s \leqslant \infty

. The modified Hardy operators

T
η

f
(
x
)
=

x

−
η

T
f
(
x
)

{T_\eta }f(x) = {x^{ - \eta }}Tf(x)

for

η
\eta

real are also treated.
The ∂̄-Neumann solution to the inhomogeneous Cauchy-Riemann equation in the ball in 𝐶ⁿ
Authors: Harvey F. Reese , Polking John C.
Published: June 2010

Let

ϑ
\vartheta

denote the formal adjoint of the Cauchy-Riemann operator

∂
¯

\overline \partial

on

C

n

{{\mathbf {C}}^n}

, and let

N
N

denote the Kohn-Neumann operator on the unit ball in

C

n

{{\mathbf {C}}^n}

. The operator

ϑ

∘

N

\vartheta \; \circ \;N

provides a natural fundamental solution for

∂
¯

f
=
g

\overline \partial f = g

on the ball. It is our purpose to present the kernel

P
P

for this operator

ϑ

∘

N

\vartheta \; \circ \;N

explicitly—the coefficients are exhibited as rational functions.
Constructions arising from Néron’s high rank curves
Authors: Fried M.
Published: June 2010

Many papers quote Néron’s geometric construction of elliptic curves of rank

11
11

over

Q

[

N

]

\mathbb {Q}\;[{\mathbf {N}}]

—still, at the writing of this paper, the elliptic curves of highest demonstrated rank. The purported reason for the ordered display of "creeping rank" in [

P
P

,

G
Z

,

N
a

{\mathbf {PP}},{\mathbf {GZ}},{\mathbf {Na}}

and

B
K

{\mathbf {BK}}

] is to make

[

N

]

[{\mathbf {N}}]

explicit. Excluding

[

B
K

]

[{\mathbf {BK}}]

, however, these papers derive little from Néron’s constructions. All show some lack of confidence in the details of

[

N

]

[{\mathbf {N}}]

. The core of this paper (

§
3

\S 3

), meets objections to

[

N

]

[{\mathbf {N}}]

raised by correspondents. Our method adds a novelty as it magnifies the constructions of

[

N

]

[{\mathbf {N}}]

—"generation of pencils of cubics from their singular fibers". This has two advantages: it displays (Remark 4.2) the free parameters whose specializations give high rank curves; and it demonstrates the existence of rank

11
11

curves through one appeal only to Hilbert’s irreducibility theorem. That is, we have eliminated the unusual analogue of Hilbert’s result that takes up most of

[

N

]

[{\mathbf {N}}]

. In particular

(
§
4
(
c
)
)

(\S 4(c))

, the explicit form of the irreducibility theorem in

[

F
r

]

[{\mathbf {Fr}}]

applies to give explicit rank

11
11

curves over

Q

\mathbb {Q}

: with Selmer’s conjecture, rank

12
12

.
The level sets of the moduli of functions of bounded characteristic
Authors: Berman Robert D.
Published: June 2010

For

f
f

a nonconstant meromorphic function on

Δ
=
{

|

z

|

>
1
}

\Delta = \{ |z| > 1\}

and

r
∈
(
inf

|

f

|

,
sup

|

f

|

)

r \in (\inf |f|,\sup |f|)

, let

L

(
f
,
r
)
=
{
z
∈
Δ
:

|

f
(
z
)

|

=
r
}

\mathcal {L}(f,r) = \{ z \in \Delta :|f(z)| = r\}

. In this paper, we study the components of

Δ
∖

L

(
f
,
r
)

\Delta \backslash \mathcal {L}(f,r)

along with the level sets

L

(
f
,
r
)

\mathcal {L}(f,r)

. Our results include the following: If

f
f

is an outer function and

Ω
\Omega

a component of

Δ
∖

L

(
f
,
r
)

\Delta \backslash \mathcal {L}(f,r)

, then

Ω
\Omega

is a simply-connected Jordan region for which

(

fr

Ω
)
∩
{

|

z

|

=
1
}

({\text {fr}}\;\Omega ) \cap \{ |z| = 1\}

has positive measure. If

f
f

and

g
g

are inner functions with

L

(
f
,
r
)
=

L

(
g
,
s
)

\mathcal {L}\,(f,r) = \mathcal {L}\,(g,s)

, then

g
=
η

f
α

g = \eta {f^\alpha }

, where

|

η

|

=
1

|\eta | = 1

and

α
>
0

\alpha > 0

. When

g
g

is an arbitrary meromorphic function, the equality of two pairs of level sets implies that

g
=
c

f
α

g = c{f^\alpha }

, where

c
≠
0

c \ne 0

and

α
∈
(
−
∞
,
∞
)

\alpha \in ( - \infty ,\infty )

. In addition, an inner function can never share a level set of its modulus with an outer function. We also give examples to demonstrate the sharpness of the main results.
Projectively equivalent metrics subject to constraints
Authors: Taber William
Published: June 2010

This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics

g
g

and

g
∗

{g^ \ast }

on a manifold

M
M

that induce the same Riemannian metric on a hypersurface

H
H

. In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set,

C
C

, of points at which

g
g

and

g
∗

{g^ \ast }

are conformally related. The space

(
M
−
C
,
g
)

(M - C,g)

is locally a warped product manifold over the hypersurface

H
H

. In the Lorentz setting,

C
C

is empty. In the Riemannian case,

C
C

contains at most two points. If

C
C

is nonempty, then

H
H

is isometric to a standard sphere. Furthermore, in the case that

C
C

contains one point, natural hypotheses imply

M
M

is diffeomorphic to

R
n

{R^n}

. If

C
C

contains two points

M
M

is diffeomorphic to

S
n

{S^n}

.
The theory of ordered abelian groups does not have the independence property
Authors: Gurevich Y. , Schmitt P. H.
Published: June 2010

We prove that no complete theory of ordered abelian groups has the independence property, thus answering a question by B. Poizat. The main tool is a result contained in the doctoral dissertation of Yuri Gurevich and also in P. H. Schmitt’s Elementary properties of ordered abelian groups, which basically transforms statements on ordered abelian groups into statements on coloured chains. We also prove that every

n
n

-type in the theory of coloured chains has at most

2
n

{2^n}

coheirs, thereby strengthening a result by B. Poizat.
Mean value properties of the Laplacian via spectral theory
Authors: Strichartz Robert S.
Published: June 2010

Let

ϕ
(

z
2

)

\phi ({z^2})

be an even entire function of temperate exponential type,

L
L

a selfadjoint realization of

−
Δ
+
c

(
x
)

- \Delta + c\,(x)

, where

Δ
\Delta

is the Laplace-Beltrami operator on a Riemannian manifold, and

ϕ

(
L
)

\phi \,(L)

the operator given by spectral theory. A Paley-Wiener theorem on the support of

ϕ

(
L
)

\phi \,(L)

is proved, and is used to show that

L
u
=
λ
u

Lu = \lambda u

on a suitable domain implies

ϕ

(
L
)

u
=
ϕ

(
λ
)

u

\phi \,(L)\,u = \phi \,(\lambda )\,u

, as well as a generalization of Àsgeirsson’s theorem. A concrete realization of the operators

ϕ

(
L
)

\phi \,(L)

is given in the case of a compact Lie group or a noncompact symmetric space with complex isometry group.
Unicellular operators
Authors: Barría José , Davidson Kenneth R.
Published: June 2010

An operator is unicellular if its lattice of invariant subspaces is totally ordered by inclusion. The list of nests which are known to be the set of invariant subspaces of a unicellular operator is surprisingly short. We construct unicellular operators on

l
p

,
1
⩽
p
>
∞

{l^p},1 \leqslant p > \infty

, and on

c
0

{c_0}

with lattices isomorphic to

α
+
X
+

β

∗

\alpha + X + {\beta ^{\ast }}

where

α
\alpha

and

β
\beta

are countable (finite or zero) ordinals, and

X
X

is in this short list. Certain other nests are attained as well.
The family approach to total cocompleteness and toposes
Authors: Street Ross
Published: June 2010

A category with small homsets is called total when its Yoneda embedding has a left adjoint; when the left adjoint preserves pullbacks, the category is called lex total. Total categories are characterized in this paper in terms of special limits and colimits which exist therein, and lex-total categories are distinguished as those which satisfy further exactness conditions. The limits involved are finite limits and intersections of all families of subobjects. The colimits are quotients of certain relations (called congruences) on families of objects (not just single objects). Just as an arrow leads to an equivalence relation on its source, a family of arrows into a given object leads to a congruence on the family of sources; in the lex-total case all congruences arise in this way and their quotients are stable under pullback. The connection with toposes is examined.
Characteristic, maximum modulus and value distribution
Authors: Hayman W. K. , Rossi J. F.
Published: June 2010

Let

f
f

be an entire function such that

log
⁡
M
(
r
,
f
)
∼
T
(
r
,
f
)

\log M(r,f)\sim T(r,f)

on a set

E
E

of positive upper density. Then

f
f

has no finite deficient values. In fact, if we assume that

E
E

has density one and

f
f

has nonzero order, then the roots of all equations

f
(
z
)
=
a

f(z) = a

are equidistributed in angles. In view of a recent result of Murai [6] the conclusions hold in particular for entire functions with Fejér gaps.
Function theoretic results for complex interpolation families of Banach spaces
Authors: Rochberg Richard
Published: June 2010

The theory of complex interpolation of Banach spaces is viewed as a branch of the theory of vector valued holomorphic functions. Versions of the Schwarz lemma, Liouville’s theorem, the identity theorem and the reflection principle are proved and are interpreted from the point of view of interpolation theory.
Difference equations, isoperimetric inequality and transience of certain random walks
Authors: Dodziuk Jozef
Published: June 2010

The difference Laplacian on a square lattice in

R

n

{{\mathbf {R}}^n}

has been studied by many authors. In this paper an analogous difference operator is studied for an arbitrary graph. It is shown that many properties of the Laplacian in the continuous setting (e.g. the maximum principle, the Harnack inequality, and Cheeger’s bound for the lowest eigenvalue) hold for this difference operator. The difference Laplacian governs the random walk on a graph, just as the Laplace operator governs the Brownian motion. As an application of the theory of the difference Laplacian, it is shown that the random walk on a class of graphs is transient.
Tensor products for the de Sitter group
Authors: Martin Robert P.
Published: June 2010

The decomposition of the tensor product of a principal series representation with any other irreducible unitary representation of

G
G

is determined for the simply connected double covering,

G
=
Spin

(
4
,
1
)

G = \operatorname {Spin}\,(4,1)

, of the DeSitter group.
Some applications of the topological characterizations of the sigma-compact spaces 𝑙²_{𝑓} and Σ
Authors: Curtis Doug , Dobrowolski Tadeusz , Mogilski Jerzy
Published: June 2010

We use a technique involving skeletoids in

σ
\sigma

-compact metric

ARs
\text {ARs}

to obtain some new examples of spaces homeomorphic to the

σ
\sigma

-compact linear spaces

l
f
2

l_f^2

and

Σ
\Sigma

. For example, we show that (1) every

ℵ
0

{\aleph _0}

-dimensional metric linear space is homeomorphic to

l
f
2

l_f^2

; (2) every

σ
\sigma

-compact metric linear space which is an

AR
\text {AR}

and which contains an infinite-dimensional compact convex subset is homeomorphic to

Σ
\Sigma

; and (3) every weak product of a sequence of

σ
\sigma

-compact metric

ARs
\text {ARs}

which contain Hilbert cubes is homeomorphic to

Σ
\Sigma

.
Convergence of multivariate polynomials interpolating on a triangular array
Authors: Goodman T. N. T. , Sharma A.
Published: June 2010

Given a triangular array of complex numbers, it is well known that for any function

f
f

smooth enough, there is a unique polynomial

G
n

f

{G_n}f

of degree

≤
n

\leq n

such that on each of the first

n
+
1

n + 1

rows of the array the divided difference of

G
n

f

{G_n}f

coincides with that of

f
f

. This result has recently been generalized to give a unique polynomial

G

n

f

{\mathcal {G}_n}f

in

k
k

variables

(
k
>
1
)

(k > 1)

of total degree

≤
n

\leq n

which interpolates a given function

f
f

on a triangular array in

C
k

{C^k}

. In this paper we extend some results of A. O. Gelfond and derive formulas for

G

n

f

{\mathcal {G}_n}f

and

f
−

G

n

f

f - {\mathcal {G}_n}f

to prove some results on convergence of

G

n

f

{\mathcal {G}_n}f

to

f
f

as

n
→
∞

n \to \infty

under various conditions on

f
f

and on the triangular array.
Boundedness of fractional operators on 𝐿^{𝑝} spaces with different weights
Authors: Harboure Eleonor , Macías Roberto A. , Segovia Carlos
Published: June 2010

Let

T
α

{T_\alpha }

be either the fractional integral operator

∫
f
(
y
)

|

x
−
y

|

α
−
n

d
y

\smallint f(y)|x - y{|^{\alpha - n}}\; dy

, or the fractional maximal operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup left-brace right-brace colon times times r minus minus alpha n integral integral greater-than greater-than vertical-bar vertical-bar minus minus xyr vertical-bar vertical-bar of ff left-parenthesis right-parenthesis y separator d separator y colon greater-than greater-than r 0">

sup

{

r

α
−
n

∫

|

x
−
y

|

>
r

|

f
(
y
)

|

d
y
:

r
>
0

}

\sup \left \{ {{r^{\alpha - n}}{\smallint _{|x - y| > r}}|f(y)|dy:\,r > 0} \right \}

. Given a weight

w
w

(resp.

υ
\upsilon

), necessary and sufficient conditions are given for the existence of a nontrivial weight

υ
\upsilon

(resp.

w
w

) such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis integral StartAbsoluteValue upper T Subscript alpha Baseline f EndAbsoluteValue Superscript q Baseline upsilon d x right-parenthesis Superscript 1 slash q Baseline less-than-or-slanted-equals left-parenthesis integral StartAbsoluteValue f EndAbsoluteValue Superscript p Baseline w d x right-parenthesis Superscript 1 slash p">

(
∫

|

T
α

f

|

q

υ

d
x

)

1

/

q

⩽

(
∫

|

f

|

p

w

d
x

)

1

/

p

{(\smallint |{T_\alpha }f{|^q}\upsilon \;dx)^{1/q}} \leqslant {(\smallint |f{|^p}w\;dx)^{1/p}}

holds. Weak type substitutes in limiting cases are considered.
Surgery on Poincaré complexes
Authors: Hodgson J. P. E.
Published: June 2010

A geometric approach to surgery on Poincaré complexes is described. The procedure mimics the original techniques for manifolds. It is shown that the obstructions to surgering a degree-one normal map of Poincaré complexes to a homotopy equivalence lie in the Wall groups, and that all elements in these groups can arise as obstructions.
Bilinear forms on 𝐻^{∞} and bounded bianalytic functions
Authors: Bourgain J.
Published: June 2010

Given an arbitrary Radon probability measure on the circle

π
\pi

, a generlization of the classical Cauchy transform is obtained. These projections are used to prove that each bounded linear operator from a reflexive subspace of

L
1

{L^1}

or

L
1

(
π
)

/

H
1

{L^1}(\pi )/{H^1}

into

H
∞

(
D
)

{H^\infty }(D)

admits a bounded extension. These facts lead to different variants of the cotype-

2
2

inequality for

L
1

(
π
)

/

H
1

{L^1}(\pi )/{H^1}

. Applications are given to absolutely summing operators and the existence of certain bounded bianalytic functions. For instance, we derive the Hilbert space factorization of arbitrary bounded linear operators from

H
∞

(
D
)

{H^\infty }(D)

into its dual without an a priori approximation hypothesis, thus completing some of the work in [1]. Our methods give new information about the Fourier coefficients of

H
∞

(
D
×
D
)

{H^\infty }(D \times D)

-functions, thus improving a theorem in [6].
Finite time analyticity for the two- and three-dimensional Rayleigh-Taylor instability
Authors: Sulem C. , Sulem P.-L.
Published: June 2010

The Rayleigh-Taylor instability refers to the dynamics of the interface between two ideal irrotational fluids of different densities superposed one over the other and in relative motion. The well-posedness of this problem is considered for two- and three-dimensional flows in the entire space and in the presence of a horizontal bottom. In the entire space, finite time analyticity of the interface is proven when the initial interface has sufficiently small gradients and is flat at infinity. In the presence of a horizontal bottom, the initial interface corrugations has also to be small initially but it is not required to vanish at infinity.
On a.e. convergence of solutions of hyperbolic equations to 𝐿^{𝑝}-initial data
Authors: Ruiz Alberto
Published: June 2010

We consider the Cauchy data problem

u
(
x
,
0
)
=
0

u(x,0) = 0

,

∂
u
(
x
,
0
)

/

∂
t
=
f
(
x
)

\partial u(x,0)/\partial t = f(x)

, for a strongly hyperbolic second order equation in

n
n

th spatial dimension,

n
≥
3

n \geq 3

, with

C
∞

{C^\infty }

coefficients. Almost everywhere convergence of the solution of this problem to initial data, in the appropriate sense is proved for

f
f

in

L
p

{L^p}

,

2
n

/

(
n
+
1
)
>
p
>
2
(
n
−
2
)

/

(
n
−
3
)

2n/(n + 1) > p > 2(n - 2)/(n - 3)

. The basic techniques are

L
p

{L^p}

-estimates for some maximal operators associated to the problem (see [4]), and the asymptotic expansion of the Riemann function given by D. Ludwig (see [9]).
Conway’s field of surreal numbers
Authors: Alling Norman L.
Published: June 2010

Conway introduced the Field

N
o

{\mathbf {No}}

of numbers, which Knuth has called the surreal numbers.

N
o

{\mathbf {No}}

is a proper class and a real-closed field, with a very high level of density, which can be described by extending Hausdorff’s

η
ξ

{\eta _\xi }

condition. In this paper the author applies a century of research on ordered sets, groups, and fields to the study of

N
o

{\mathbf {No}}

. In the process, a tower of subfields,

ξ

N
o

\xi {\mathbf {No}}

, is defined, each of which is a real-closed subfield of

N
o

{\mathbf {No}}

that is an

η
ξ

{\eta _\xi }

-set. These fields all have Conway partitions. This structure allows the author to prove that every pseudo-convergent sequence in

N
o

{\mathbf {No}}

has a unique limit in

N
o

{\mathbf {No}}

.
Spaces with coarser minimal Hausdorff topologies
Authors: Porter Jack , Vermeer Johannes
Published: June 2010

A technique is developed, using

H
H

-closed extensions, for determining when certain Hausdorff spaces are Katetov, i.e., have a coarser minimal Hausdorff topology. Our technique works for Čech-complete Lindelöf spaces, complete metrizable spaces, and many other spaces. Also, a number of interesting examples are presented; the most striking is an example of a Katetov space whose semiregularization is not Katetov.
A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc
Authors: Row W. H. , Walsh John J.
Published: June 2010

A decomposition

G
G

of

S
3

{S^3}

is constructed with the following properties: (1) The set

N
G

{N_G}

of all nondegenerate elements consists of a null sequence of arcs and

J
=

CL

(
∪
{
g
∈

N
G

}
)

J = {\text {CL}}( \cup \{ g \in {N_G}\} )

is a simple closed curve. (2) Each arc contained in

J
J

is cellular. (3)

J
J

is the boundary of a disk

Q
Q

that is locally flat except at points of

J
J

. (4) The decomposition

G
G

is not shrinkable; that is, the decomposition space is not homeomorphic to

S
3

{S^3}

.
The fine structure of transitive Riemannian isometry groups. I
Authors: Gordon Carolyn S. , Wilson Edward N.
Published: June 2010

Let

M
M

be a connected homogeneous Riemannian manifold,

G
G

the identity component of the full isometry group of

M
M

and

H
H

a transitive connected subgroup of

G
G

.

G
=
H
L

G = HL

, where

L
L

is the isotropy group at some point of

M
M

.

M
M

is naturally identified with the homogeneous space

H

/

H
∩
L

H/H \cap L

endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of

M
M

as a Riemannian homogeneous space of a connected Lie group

H
H

, describe the structure of the full connected isometry group

G
G

in terms of

H
H

. This problem has already been studied in case

H
H

is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.
Ergodic semigroups of epimorphisms
Authors: Berend Daniel
Published: June 2010

The conditions for ergodicity of semigroups of epimorphisms of compact groups are studied. In certain cases ergodic semigroups are shown to contain small ergodic subsemigroups. Properties related to ergodicity, such as that of admitting no infinite closed invariant proper subset of the group, are discussed for semigroups of epimorphisms and of affine transformations.
Fractional integrals on weighted 𝐻^{𝑝} spaces
Authors: Gatto Angel E. , Gutiérrez Cristian E. , Wheeden Richard L.
Published: June 2010

We characterize the pairs of doubling weights

(
u
,
v
)

(u,v)

on

R
n

{R^n}

such that
\[

∥

I
α

f

∥

H
u
q

≤
c
∥
f

∥

H
v
p

,

0
>
p
>
q
>
∞

\parallel {I_\alpha }f{\parallel _{H_u^q}} \leq c\parallel f{\parallel _{H_v^p}}, \quad 0 > p > q > \infty

\]
, where

I
α

,
α
>
0

{I_\alpha },\alpha > 0

, is the fractional integral operator. We also consider the behavior of an associated maximal function. Applications of the results to Sobolev inequalities in weighted

L
p

{L^p}

spaces are given.
Convolution equations in spaces of distributions with one-sided bounded support
Authors: Shambayati R. , Zielezny Z.
Published: June 2010

Let

D

′
(
0
,
∞
)

\mathcal {D}\prime (0,\infty )

be the space of distributions on

R
R

with support in

[
0
,
∞
)

[0,\infty )

and

S

′
(
0
,
∞
)

\mathcal {S}\prime (0,\infty )

its subspace consisting of tempered distributions. We characterize the distributions

S
∈

D

′
(
0
,
∞
)

S \in \mathcal {D}\prime (0,\infty )

for which

S

∗

D

′
(
0
,
∞
)
=

D

′
(
0
,
∞
)

S\, \ast \mathcal {D}\prime (0,\infty ) = \mathcal {D}\prime (0,\infty )

, where

∗
\ast

is the convolution. We also characterize the distributions

S
∈

S

′
(
0
,
∞
)

S \in \mathcal {S}\prime (0,\infty )

for which

S
∗

S

′
(
0
,
∞
)
=

S

′
(
0
,
∞
)

S \ast \mathcal {S}\prime (0,\infty ) = \mathcal {S}\prime (0,\infty )

.
On a class of Banach spaces of functions associated with the notion of entropy
Authors: Korenblum Boris
Published: June 2010

A class of function spaces on the circle is introduced which contain all continuous functions of bounded variation but are included in the set of all continuous functions. The corresponding dual spaces consist of certain types of generalized measures. One application of these spaces is a new convergence test for Fourier series which includes both the Dirichlet-Jordan and the Dini-Lipschitz tests.
Propagation of singularities for nonstrictly hyperbolic semilinear systems in one space dimension
Authors: Micheli Lucio
Published: June 2010

We consider the creation and propagation of singularities in the solutions of semilinear nonstrictly hyperbolic systems in one space dimension when the initial data has jump discontinuities. We show that singularities travelling along characteristics can branch at points of degeneracy of the vector fields on all other forward characteristics. We prove a lower bound for the strength of these new singularities, and we give an example showing that our result cannot be improved in general.
On the solution of certain systems of partial difference equations and linear dependence of translates of box splines
Authors: Dahmen Wolfgang , Micchelli Charles A.
Published: June 2010

This paper is concerned with solving systems of partial difference equations which arise when studying linear dependence of translates of box splines.
𝐿^{𝑝} estimates for Schrödinger evolution equations
Authors: Balabane M. , Emamirad H. A.
Published: June 2010

We prove that for Cauchy data in

L
1

(

R

n

)

{L^1}({{\mathbf {R}}^n})

, the solution of a Schrödinger evolution equation with constant coefficients of order

2
m

2m

is uniformly bounded for

t
≠
0

t \ne 0

, with bound

(
1
+

|

t

|

−
c

)

(1 + |t{|^{ - c}})

, where

c
c

is an integer,

c
>
n

/

2
m
−
1

c > n/2m - 1

. Moreover it belongs to

L
q

(

R

n

)

{L^q}({{\mathbf {R}}^n})

if

q
>
q
(
m
,
n
)

q > q(m,n)

, with its

L
p

{L^p}

norm bounded by

(

|

t

|

c
′

+

|

t

|

−
c

)

(|t{|^{c’}} + |t{|^{ - c}})

, where

c
′

c’

is an integer,

c
′

>
n

/

q

c’ > n/q

. A maximal local decay result is proved. Interpolating between

L
1

{L^1}

and

L
2

{L^2}

, we derive

(

L
p

,

L
q

)

({L^p},{L^q})

estimates. On the other hand, we prove that for Cauchy data in

L
p

(

R

n

)

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

, such a Cauchy problem is well posed as a distribution in the

t
t

-variable with values in

L
p

(

R

n

)

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

, and we compute the order of the distribution. We apply these two results to the study of Schrödinger equations with potential in

L
p

(

R

n

)

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

. We give an estimate of the resolvent operator in that case, and prove an asymptotic boundedness for the solution when the Cauchy data belongs to a subspace of

L
p

(

R

n

)

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

.
Stationary solutions of chemotaxis systems
Authors: Schaaf Renate
Published: June 2010

The Keller-Segel Model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance (Chemotaxis). This paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns. Examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response.
Conjugacy problem in 𝐺𝐿₂(𝑍[√-1]) and units of quadratic extensions of 𝑄(√-1)
Authors: Onishi Hironori
Published: June 2010

A highly efficient procedure for deciding if two given elements of

G

L

2

(

Z

[

−
1

]
)

{\text {G}}{{\text {L}}_2}(\mathbf {Z}[\sqrt { - 1} ])

are conjugate or not will be presented. It makes use of a continued fraction algorithm in

Z

[

−
1

]

\mathbf {Z}[\sqrt { - 1} ]

and gives a fundamental unit of any given quadratic extension of

Q

(

−
1

)

\mathbf {Q}(\sqrt { - 1} )

.
The fifth and seventh order mock theta functions
Authors: Andrews George E.
Published: June 2010

The theory of Bailey chains is extended to yield identities for Hecke type modular forms and related generalizations. The extended results allow us to produce Hecke type series for the fifth and seventh order mock theta functions. New results on the generating function for sums of three squares also follow, and a new proof that every integer is the sum of three triangular numbers is given.
On periodic solutions of superlinear parabolic problems
Authors: Esteban Maria J.
Published: June 2010

In this paper we study the existence of positive nontrivial periodic solutions of semilinear parabolic problems. Most of the nonlinearities considered are of the superlinear type. Some bifurcation results are proved as well.
Baer’s lemma and Fuchs’s problem 84a
Authors: Albrecht Ulrich
Published: June 2010

An indecomposable, torsion-free, reduced abelian group

A
A

has the properties that (i) each subgroup

B
B

of an

A
A

-projective group with

S
A

(
B
)
=
B

{S_A}(B) = B

is

A
A

-projective and (ii) each subgroup

B
B

of a group

G
G

with

S
A

(
G
)
+
B
=
G

{S_A}(G) + B = G

and

G

/

B

G/B

A
A

-projective is a direct summand if and only if

A
A

is self-small and flat as a left

E
(
A
)

E(A)

-module, and

E
(
A
)

E(A)

is right hereditary. Furthermore, a group-theoretic characterization is given for torsion-free, reduced abelian groups with a right and left Noetherian, hereditary endomorphism ring. This is applied to Fuchs’ Problem 84a. Finally, various applications of the results of this paper are given.
Entire functions which are infinitely integer-valued at a finite number of points
Authors: Lockhart P. , Straus E. G.
Published: June 2010

This paper determines arithmetic limits for the growth rates of entire functions which are infinitely integer valued on a finite set

S
S

. The characterization of such functions with growth rate less than the arithmetic limit is complete if there exist exponential polynomials which are infinitely integer valued on

S
S

.
On the dichotomy problem for tensor algebras
Authors: Bourgain J.
Published: June 2010

Let

I
I

,

J
J

be discrete spaces and

E
⊂
I
×
J

E \subset I \times J

. Then either

E
E

is a

V
V

-Sidon set (in the sense of

[

2

,
§
11
]

[{\mathbf {2}},\S 11]

), or the restriction algebra

A
(
E
)

A(E)

is analytic. The proof is based on probabilistic methods, involving Slépian’s lemma.
On multiplication algebras
Authors: Finston David R.
Published: June 2010

The basic properties of multiplication algebras of nonassociative algebras over rings are introduced, including a discussion of multiplication algebras of tensor products of algebras. A characterization of semisimple artinian multiplication algebras is given along with a discussion of the nature of the simple factors of a multiplication algebra modulo its Jacobson radical. A criterion distinguishing the multiplication algebras of certain associative algebras is proved. Examples are included to illustrate certain proved results.
The distribution of solutions to equations over finite fields
Authors: Cochrane Todd
Published: June 2010

Let

F

q

{\mathbb {F}_q}

be the finite field in

q
=

p
f

q = {p^f}

elements,

F
_

(

x
_

)

\underline F (\underline x )

be a

k
k

-tuple of polynomials in

F

q

[

x
1

,
…
,

x
n

]

{\mathbb {F}_q}[{x_1}, \ldots ,{x_n}]

,

V
V

be the set of points in

F

q
n

\mathbb {F}_q^n

satisfying

F
_

(

x
_

)
=

0
_

\underline F (\underline x ) = \underline 0

and

S
S

,

T
T

be any subsets of

F

q
n

\mathbb {F}_q^n

. Set

ϕ
(
V
,

0
_

)
=

|

V

|

−

q

n
−
k

\phi (V,\underline 0 ) = |V| - {q^{n - k}}

,
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi left-parenthesis upper V comma y underbar right-parenthesis equals sigma-summation Underscript x underbar element-of upper V Endscripts e left-parenthesis StartFraction 2 pi i Over p EndFraction trace left-parenthesis x underbar dot y underbar right-parenthesis right-parenthesis for y underbar not-equals ModifyingBelow 0 With bar comma">

ϕ
(
V
,

y
_

)
=

∑

x
_

∈
V

e

(

2
π
i

p

Tr
⁡
(

x
_

⋅

y
_

)

)

for

y
_

≠

0
_

,

\phi (V,\underline y ) = \sum \limits _{\underline x \in V} {e\left ( {\frac {{2\pi i}} {p}\operatorname {Tr} (\underline x \cdot \underline y )} \right )\quad {\text {for}}\;\underline y \ne \underline 0 ,}

\]
and

Φ
(
V
)
=

max

y
_

|

ϕ
(
V
,

y
_

)

|

\Phi (V) = {\max _{\underline y }}|\phi (V,\underline y )|

. We use finite Fourier series to show that

(
S
+
T
)
∩
V

(S + T) \cap V

is nonempty if

|

S

|

|

T

|

>

Φ
2

(
V
)

q

2
k

|S||T| > {\Phi ^2}(V){q^{2k}}

. In case

q
=
p

q = p

we deduce from this, for example, that if

C
C

is a convex subset of

R

n

{\mathbb {R}^n}

symmetric about a point in

Z

n

{\mathbb {Z}^n}

, of diameter

>
2
p

> 2p

(with respect to the sup norm), and

Vol
⁡
(
C
)
>

2

2
n

Φ
(
V
)

p
k

\operatorname {Vol} (C) > {2^{2n}}\Phi (V){p^k}

, then

C
C

contains a solution of

F
_

(

x
_

)
≡

0
_

(
mod
p
)

\underline F (\underline x ) \equiv \underline 0 (\bmod p)

. We also show that if

B
B

is a box of points in

F

q
n

\mathbb {F}_q^n

not contained in any

(
n
−
1
)

(n - 1)

-dimensional subspace and

|

B

|

>
4
⋅

2

n
f

Φ
(
V
)

q
k

|B| > 4 \cdot {2^{nf}}\Phi (V){q^k}

, then

B
∩
V

B \cap V

contains

n
n

linearly independent points.
Some generalized Brown-Gitler spectra
Authors: Goerss Paul G. , Jones John D. S. , Mahowald Mark E.
Published: June 2010

Brown-Gitler spectra for the homology theories associated with the spectra

K

Z

p

∧

K{{\mathbf {Z}}_p}^ \wedge

,

b
o

bo

, and

b
u

bu

are constructed. Complexes adapted to the new Brown-Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.
A bilaterally deterministic 𝜌-mixing stationary random sequence
Authors: Bradley Richard C.
Published: June 2010

A (nondegenerate) strictly stationary sequence

(

X
k

,

k
∈

Z

)

({X_k},\;k \in {\mathbf {Z}})

of random variables is constructed such that the

ρ
\rho

-mixing (maximal correlation mixing) condition is satisfied and each

X
k

{X_k}

is measurable with respect to the double tail

σ
\sigma

-field.