bluuebunny/crossref_metadata_2025 huggingface dataset:
Derived functors of unitary highest weight modules at reduction points
Authors: Frajria Pierluigi Möseneder
Published: June 2012

The derived functors introduced by Zuckerman are applied to the unitary highest weight modules of the Hermitian symmetric pairs of classical type. The construction yields "small" unitary representations which do not have a highest weight. The infinitesimal character parameter of the modules we consider is such that their derived functors are nontrivial in more than one degree; at the extreme degrees where the cohomology is nonvanishing, it is possible to determine the

K

{\mathbf {K}}

-spectrum of the resulting representations completely. Using this information it is shown that, in most cases, the derived functor modules are unitary, irreducible, and not of highest weight type. Their infinitesimal character and lowest

K

{\mathbf {K}}

-type are also easily computed.
Weak type estimates for a singular convolution operator on the Heisenberg group
Authors: Grafakos Loukas
Published: June 2012

On the Heisenberg group

H

n

{\mathbb {H}^n}

with coordinates

(
z
,
t
)
∈

C

n

×

R

(z,t) \in {\mathbb {C}^n} \times \mathbb {R}

, define the distribution

K
(
z
,
t
)
=
L
(
z
)
δ
(
t
)

K(z,t) = L(z)\delta (t)

, where

L
(
z
)

L(z)

is a homogeneous distribution on

C

n

{\mathbb {C}^n}

of degree

−
2
n

- 2n

, smooth away from the origin and

δ
(
t
)

\delta (t)

is the Dirac mass in the

t
t

variable. We prove that the operator given by convolution with

K
K

maps

H
1

(

H

n

)

{H^1}({\mathbb {H}^n})

to weak

L
1

(

H

n

)

{L^1}({\mathbb {H}^n})

.
A modified Schur algorithm and an extended Hamburger moment problem
Authors: Njåstad Olav
Published: June 2012

An algorithm for a Pick-Nevanlinna problem where the interpolation points coalesce into a finite set of points on the real line is introduced, its connection with certain multipoint Padé approximation problems is discussed, and the results are used to obtain the solutions of an extended Hamburger moment problem.
Asymptotic integrations of nonoscillatory second order differential equations
Authors: Chen Shao Zhu
Published: June 2012

The linear differential equation (1)

(
r
(
t
)

x
′

)
′

+
(
f
(
t
)
+
q
(
t
)
)
x
=
0

(r(t)x’)’ + (f(t) + q(t))x= 0

is viewed as a perturbation of the equation (2)

(
r
(
t
)

y
′

)
′

+
(
f
(
t
)
y
=
0

(r(t)y’)’ + (f(t)y = 0

, where

r
>
0

r > 0

,

f
f

and

q
q

are real-valued continuous functions. Suppose that (2) is nonoscillatory at infinity and

y
1

{y_1}

,

y
2

{y_2}

are principal, nonprincipal solutions of (2), respectively. Adapted Riccati techniques are used to obtain an asymptotic integration for the principal solution

x
1

{x_1}

of (1). Under some mild assumptions, we characterize that (1) has a principal solution

x
1

{x_1}

satisfying

x
1

=

y
1

(
1
+
o
(
1
)
)

{x_1}= {y_1}(1 + o(1))

. Sufficient (sometimes necessary) conditions under which the nonprincipal solution

x
2

{x_2}

of (1) behaves, in three different degrees, like

y
2

{y_2}

as

t
→
∞

t \to \infty

are also established.
Parity and generalized multiplicity
Authors: Fitzpatrick P. M. , Pejsachowicz Jacobo
Published: June 2012

Assuming that

X
X

and

Y
Y

are Banach spaces and

α
:
[
a
,
b
]
→

L

(
X
,
Y
)

\alpha :[a,b] \to \mathcal {L}(X,Y)

is a path of linear Fredholm operators with invertible endpoints, in

[

F

−
P
1
]

[{\text {F}} - \text {P}1]

we defined a homotopy invariant of

α
,
σ
(
α
,
I
)
∈

Z

2

\alpha ,\sigma (\alpha ,I) \in {{\mathbf {Z}}_2}

, the parity of

α
\alpha

on

I
I

. The parity plays a fundamental role in bifurcation problems, and in degree theory for nonlinear Fredholm-type mappings. Here we prove (a) that, generically, the parity is a

mod

2

\bmod \, 2

count of the number of transversal intersections of

α
(
I
)

\alpha (I)

with the set of singular operators, (b) that if

λ
0

{\lambda _0}

is an isolated singular point of

α
\alpha

, then the local parity
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma left-parenthesis alpha comma lamda 0 right-parenthesis identical-to limit Underscript epsilon right-arrow 0 Endscripts sigma left-parenthesis alpha comma left-bracket lamda 0 minus epsilon comma lamda 0 plus epsilon right-bracket right-parenthesis">

σ
(
α
,

λ
0

)
≡

lim

ε
→
0

σ
(
α
,
[

λ
0

−
ε
,

λ
0

+
ε
]
)

\sigma (\alpha ,{\lambda _0}) \equiv \lim \limits _{\varepsilon \to 0} \sigma (\alpha ,[{\lambda _0} - \varepsilon ,{\lambda _0} + \varepsilon ])

\]
remains invariant under Lyapunov-Schmidt reduction, and (c) that

σ
(
α
,

λ
0

)
=

(
−
1

)

M
G

(

λ
0

)

\sigma (\alpha ,{\lambda _0}) = {(- 1)^{{M_G}({\lambda _0})}}

, where

M
G

(

λ
0

)

{M_G}({\lambda _0})

is any one of the various concepts of generalized multiplicity which have been defined in the context of linearized bifurcation data.
On complete congruence lattices of complete lattices
Authors: Grätzer G. , Lakser H.
Published: June 2012

The lattice of all complete congruence relations of a complete lattice is itself a complete lattice. In this paper, we characterize this lattice as a complete lattice. In other words, for a complete latticeLL, we construct a complete latticeKKsuch thatLLis isomorphic to the lattice of complete congruence relations ofKK. RegardingKKas an infinitary algebra, this result strengthens the characterization theorem of congruence lattices of infinitary algebras of G. Grätzer and W. A. Lampe. In addition, we show how to constructKKso that it will also have a prescribed automorphism group.
Cavitational flows and global injectivity of conformal maps
Authors: Lanza de Cristoforis Massimo
Published: June 2012

This paper treats some new mathematical aspects of the two-dimensional cavitational problem of the flow of a perfect fluid past an obstacle. Natural regularity conditions of very general type are found to ensure the global injectivity of the complex-potential and the presence of at most one zero of its derivative on the boundary of the flow. This derivative is the complex velocity. Previous authors have hypothesized the properties obtained here. The same regularity conditions are then shown to be satisfied by the classical solutions found via Villat’s integral equation. A simple counterexample in

§
4

\S 4

shows that the global injectivity of a holomorphic map defined on an unbounded Jordan domain cannot be deduced solely from its injectivity on the boundary. This simple fact raises new questions on the relation between cavitational flows and Villat’s integral equation, which are discussed in

§
3

\S 3

.
Smooth dynamics on Weierstrass nowhere differentiable curves
Authors: Hunt Brian R. , Yorke James A.
Published: June 2012

We consider a family of smooth maps on an infinite cylinder which have invariant curves that are nowhere smooth. Most points on such a curve are buried deep within its spiked structure, and the outermost exposed points of the curve constitute an invariant subset which we call the "facade" of the curve. We find that for surprisingly many of the maps in the family, all points in the facades of their invariant curves are eventually periodic.
Control of degenerate diffusions in 𝑅^{𝑑}
Authors: Hijab Omar
Published: June 2012

An optimal regularity result is established for the viscosity solution of the degenerate elliptic equation
\[

−
A
v
+
F
(
x
,
υ
,
D
υ
)
=
0
,

- Av + F(x,\upsilon ,D\upsilon )= 0,

\]

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals one half sigma-summation a Subscript i j Baseline left-parenthesis x right-parenthesis partial-differential squared slash partial-differential x Subscript i Baseline partial-differential x Subscript j Baseline comma x element-of bold upper R Superscript d">

A
=

1
2

∑

a

i
j

(
x
)

∂
2

/

∂

x

i

∂

x
j

,
x
∈

R

d

A= \frac {1}{2}\sum {{a_{ij}}(x){\partial ^2}/\partial \,x{_i}\,\partial \,{x_j}}, x \in {{\mathbf {R}}^d}

. We assume the equation is of Bellman type, i.e.
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma upsilon comma p right-parenthesis equals sup Underscript u element-of upper U Endscripts left-bracket b left-parenthesis x comma u right-parenthesis dot p plus c left-parenthesis x comma u right-parenthesis upsilon minus f left-parenthesis x comma u right-parenthesis right-bracket">

F
(
x
,
υ
,
p
)
=

sup

u
∈
U

[
b
(
x
,
u
)
⋅
p
+
c
(
x
,
u
)
υ
−
f
(
x
,
u
)
]

F(x,\upsilon ,p)= {\sup _{u \in U}}[b(x,u) \cdot p + c(x,u)\upsilon - f(x,u)]

,

U
⊂

R

d

U \subset {{\mathbf {R}}^d}

. If we set

λ
≡

inf

x
,
u

c
(
x
,
u
)

\lambda \equiv {\inf _{x,u}}c(x,u)

, then there exists

λ
0

≥
0

{\lambda _0} \geq 0

such that

0
>
λ
>

λ
0

0 > \lambda > {\lambda _0}

implies

υ
\upsilon

is Hölder, while

λ
>

λ
0

\lambda > {\lambda _0}

implies

υ
\upsilon

is Lipschitz. The following is established: Suppose the equation is also of Lipschitz type, i.e. suppose there is a Lipschitz function

u
(
x
,
υ
,
p
)

u(x,\upsilon ,p)

such that the supremum in

F

(
x
,
υ
,
p
)

F\,(x,\upsilon ,p)

is uniquely attained at

u
=
u

(
x
,
υ
,
p
)

u= u\,(x,\upsilon ,p)

; then there exists

λ
1

>

λ
0

{\lambda _1} > {\lambda _0}

such that

λ
>

λ
1

\lambda > {\lambda _1}

implies

υ
\upsilon

is

C

1
,
1

,

{C^{1,1}},

i.e.

D
υ

D\upsilon

exists and is Lipschitz.
Radon-Nikodým properties associated with subsets of countable discrete abelian groups
Authors: Dowling Patrick N.
Published: June 2012

With any subset of a countable discrete abelian we associate with it three Banach space properties. These properties are Radon-Nikodym type properties. The relationship between these properties is investigated. The results are applied to give new characterizations of Riesz subsets and Rosenthal subsets of countable discrete abelian groups.
On weak convergence in dynamical systems to self-similar processes with spectral representation
Authors: Lacey Michael T.
Published: June 2012

Let

(
X
,
μ
,
T
)

(X,\mu ,T)

be an aperiodic dynamical system. Set

S
m

f
=
f
+
⋯
+
f
∘

T

m
−
1

{S_m}f = f + \cdots + f \circ {T^{m - 1}}

, where

f
f

is a measurable function on

X
X

. Let

Y
(
t
)

Y(t)

be one of a class of self-similar process with a “nice” spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an

f
f

on

X
X

, and constants

A
m

→
+
∞

{A_m} \to + \infty

so that
\[

A
m

−
1

S

[
m
t
]

f

⇒

d

Y
(
t
)
,

A_m^{ - 1}{S_{[mt]}}f \stackrel {d}{\Rightarrow } Y(t),

\]
the convergence being understood in the sense of weak convergence of all finite dimensional distributions in

t
t

.
The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type
Authors: Boggess A. , Dwilewicz R. , Nagel A.
Published: June 2012

We show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in

C
n

{C^n}

contains an open set in

C
n

{C^n}

which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball. In addition, we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions.
Quasidiagonality of direct sums of weighted shifts
Authors: Narayan Sivaram K.
Published: June 2012

Let

H

\mathcal {H}

be a separable Hilbert space. A bounded linear operator

A
A

defined on

H

\mathcal {H}

is said to be quasidiagonal if there exists a sequence

{

P
n

}

\{ {P_n}\}

of projections of finite rank such that

P
n

→
I

{P_n} \to I

strongly and

‖
A

P
n

−

P
n

A
‖

→
0

\left \| A{P_n} - {P_n}A\right \| \to 0

as

n
→
∞

n \to \infty

. We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a

(
0
)

(0)

,

(
+
)

( + )

or

(
−
)

( - )

attached to its vertices) that can be associated with the direct sum.
On the genus of smooth 4-manifolds
Authors: Cavicchioli Alberto
Published: June 2012

The projective complex plane and the "twisted"

S
3

{S^3}

bundle over

S
1

{S^1}

are proved to be the unique closed prime connected (smooth or

PL
\text {PL}

)

4
4

-manifolds of genus two. Then the classification of the nonorientable

4
4

-manifolds of genus

4
4

is given. Finally the genus of a manifold

M
M

is shown to be related with the

2
2

nd Betti number of

M
M

and some applications are proved in the general (resp. simply-connected) case.
Positive solutions of semilinear equations in cones
Authors: Egnell Henrik
Published: June 2012

In this paper we consider the problem of finding a positive solution of the equation

Δ
u
+

|

x

|

ν

u

(
n
+
2
+
2
ν
)

/

(
n
−
2
)

=
0

\Delta u + |x{|^\nu }{u^{(n + 2 + 2\nu )/(n - 2)}} = 0

in a cone

C

\mathcal {C}

, with zero boundary data. We are only interested in solutions that are regular at infinity (i.e. such that

u
(
x
)
=
o
(

|

x

|

2
−
n

)

u(x) = o(|x{|^{2 - n}})

, as

C

∋
x
→
∞

\mathcal {C} \ni x \to \infty

). We will always assume that

ν
>
−
2

\nu > - 2

. We show that the existence of a solution depends on the sign of

ν
\nu

and also on the shape of the cone

C

\mathcal {C}

.
A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three
Authors: Cannon J. W. , Cooper Daryl
Published: June 2012

We show that a cocompact hyperbolic group in dimension

3
3

is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finite-volume hyperbolic groups in dimension

3
3

.
On the resolution of a curve lying on a smooth cubic surface in 𝑃³
Authors: Giuffrida Salvatore , Maggioni Renato
Published: June 2012

Let

C
C

be any reduced and irreducible curve lying on a smooth cubic surface in

P

3

{\mathbb {P}^3}

. In this paper we determine the graded Betti numbers of the ideal sheaf

J

C

{\mathcal {J}_C}

.
A general condition for lifting theorems
Authors: Robinson E. Arthur
Published: June 2012

We define a general condition, called stability on extensions

T
T

of measure preserving transformations

S
S

. Stability is defined in terms of relative unique ergodicity, and as a joining property. Ergodic compact group extensions are stable, and moreover stable extensions satisfy lifting theorems similar to those satisfied by group extensions. In general, stable extensions have relative entropy zero. In the class of continuous flow extensions over strictly ergodic homeomorphisms, stable extensions are generic.
The moment map of a Lie group representation
Authors: Wildberger N. J.
Published: June 2012

Given an

m
×
m

m \times m

Hadamard matrix one can extract

m
2

{m^2}

symmetric designs on

m
−
1

m - 1

points each of which extends uniquely to a

3
3

-design. Further, when

m
m

is a square, certain Hadamard matrices yield symmetric designs on

m
m

points. We study these, and other classes of designs associated with Hadamard matrices, using the tools of algebraic coding theory and the customary association of linear codes with designs. This leads naturally to the notion, defined for any prime

p
p

, of

p
p

-equivalence for Hadamard matrices for which the standard equivalence of Hadamard matrices is, in general, a refinement: for example, the sixty

24
×
24

24 \times 24

matrices fall into only six

2
2

-equivalence classes. In the

16
×
16

16 \times 16

case,

2
2

-equivalence is identical to the standard equivalence, but our results illuminate this case also, explaining why only the Sylvester matrix can be obtained from a difference set in an elementary abelian

2
2

-group, why two of the matrices cannot be obtained from a symmetric design on

16
16

points, and how the various designs may be viewed through the lens of the four-dimensional affine space over the two-element field.
The kernel-trace approach to right congruences on an inverse semigroup
Authors: Petrich Mario , Rankin Stuart
Published: June 2012

A kernel-trace description of right congruences on an inverse semigroup is developed. It is shown that the trace mapping is a complete

∩
\cap

homomorphism but not a

∨
\vee

-homomorphism. However, the trace classes are intervals in the complete lattice of right congruences. In contrast, each kernel class has a maximum element, namely the principal right congruence on the kernel, but in general there is no minimum element in a kernel class. The kernel mapping preserves neither intersections nor joins. The set of axioms presented in [7] for right kernel systems is reviewed. A new set of axioms is obtained as a consequence of the fact that a right congruence is the intersection of the principal right congruences on the idempotent classes. Finally, it is shown that even though a congruence on a regular semigroup is the intersection of the principal congruences on the idempotent classes, the situation is not the same for right congruences on a regular semigroup. Right congruences on a regular, even orthodox, semigroup are not, in general, determined by their idempotent classes.
L’espace des arcs d’une surface
Authors: Cauty Robert
Published: June 2012

We prove that, for any surface

M
M

, the space of arcs contained in

M
M

, with the topology induced by the Hausdorff distance, is homeomorphic to

M
×

∑

∞

M \times {\sum }^\infty

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation equals left-brace left-parenthesis x Subscript i Baseline right-parenthesis element-of l squared slash sigma-summation Underscript i equals 1 Overscript normal infinity Endscripts left-parenthesis i x Subscript i Baseline right-parenthesis squared greater-than normal infinity right-brace">

∑
=
{
(

x
i

)
∈

l
2

/

∑

i
=
1

∞

(
i

x
i

)

2

>
∞
}

\sum = \{ ({x_i}) \in {l^2}/\sum \nolimits _{i = 1}^\infty {{{(i{x_i})}^2} > \infty \} }

.
Isotropy groups of homotopy classes of maps
Authors: Triantafillou G.
Published: June 2012

Let

aut
⁡
(
X
)

\operatorname {aut}(X)

be the group of homotopy classes of self-homotopy equivalences of a space

X
X

and let

[
f
]
∈
[
X
,
Y
]

[f] \in [X,Y]

be a homotopy class of maps from

X
X

to

Y
Y

. The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group

aut
⁡

(
X

)

[
f
]

\operatorname {aut}{(X)_{[f]}}

of

[
f
]

[f]

under the action of

aut
⁡
(
X
)

\operatorname {aut}(X)

on

[
X
,
Y
]

[X,Y]

is commensurable to an arithmetic group. Therefore

aut
⁡

(
X

)

[
f
]

\operatorname {aut}{(X)_{[f]}}

is a finitely presented group by a result of Borel and Harish-Chandra.
A law of large numbers for fast price adjustment
Authors: Keisler H. Jerome
Published: June 2012

The purpose of this paper is to prove a law of large numbers for certain Markov processes involving large sets of weakly interacting particles. Consider a large finite set

A
A

of "particles" which move about in

m
m

-dimensional Euclidean space

R
m

{R^m}

. The particles interact with each other indirectly by means of an auxiliary quantity

p
p

in

d
d

-dimensional Euclidean space

R
d

{R^d}

. At each time

t
t

, a particle

a
∈
A

a \in A

is randomly selected and randomly jumps to a new location in

R
m

{R^m}

with a distribution depending on

p
p

and its old location. At the same time, the value of

p
p

changes to a new value depending on these same arguments. The parameter

p
p

moves by a small amount at each time but moves fast compared to the average position of the particles. Under appropriate hypotheses on the rules of motion, we shall prove the following law of large numbers. For sufficiently large

A
A

, the value of

p
p

will be close to its expected value with large probability, and the average position of the particles will be close to its expected value with large probability. The work was motivated by the problem of modelling the adjustment of prices in mathematical economics, where the particles

a
∈
A

a \in A

are agents in an exchange economy, the position of

a
a

at time

t
t

is the commodity vector held by agent

a
a

at time

t
t

, and

p
p

is the price vector at time

t
t

.
Nonnegatively curved submanifolds in codimension two
Authors: Noronha Maria Helena
Published: June 2012

Let

M
M

be a complete noncompact manifold with nonnegative sectional curvatures isometrically immersed in Euclidean spaces with codimension two. We show that

M
M

is a product over its soul, except when the soul is the circle

S
1

{S^1}

or

M
M

is

3
3

-dimensional and the soul is the Real Projective Plane. We also give a rather complete description of the immersion, including the exceptional cases.
Finite semilattices whose monoids of endomorphisms are regular
Authors: Adams M. E. , Gould Matthew
Published: June 2012

A classification is obtained for the finite semilattices

S
S

such that the monoid of endomorphisms of

S
S

is regular in the semigroup-theoretic sense.
Loewy series of certain indecomposable modules for Frobenius subgroups
Authors: Lin Zong Zhu
Published: June 2012

We imitate some approaches in infinite dimensional representation theory of complex semisimple Lie algebras by using the truncated category method in the categories of modules for certain Frobenius subgroups of a semisimple algebraic group over an algebraically closed field of characteristic

p
>
0

p > 0

. By studying the translation functors from

p
p

-singular weights to

p
p

-regular weights, we obtain some results on Loewy series of certain indecomposable modules.
Removing point singularities of Riemannian manifolds
Authors: Smith P. D. , Yang Deane
Published: June 2012

We study the behavior of geodesics passing through a point singularity of a Riemannian manifold. In particular, we show that if the curvature does not blow up too rapidly near the singularity, then the singularity is at worst an orbifold singularity. The idea is to construct the exponential map centered at a singularity. Since there is no tangent space at the singularity, a surrogate is needed. We show that the vector space of radially parallel vector fields is well defined and that there is a correspondence between unit radially parallel vector fields and geodesics emanating from the singular point.
𝐿^{𝑝} estimates for the X-ray transform restricted to line complexes of Kirillov type
Authors: Wang Hann Tzong
Published: June 2012

Let there be given a piecewise continuous rectifiable curve

ϕ
:

R

→

R

n

\phi :{\mathbf {R}} \to {{\mathbf {R}}^n}

. Let

G

1
,
n

(

M

1
,
n

)

{G_{1,n}}({M_{1,n}})

be the usual Grassmannian (bundle) in

R

n

{{\mathbf {R}}^n}

. Define an

n
n

-dimensional submanifold

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

of

M

1
,
n

{M_{1,n}}

as the set of all copies of

G

1
,
n

{G_{1,n}}

along the curve

ϕ
\phi

. Following Kirillov, we know that a nice function

f
(
x
)

f(x)

can be recovered from its X-ray transform

R

1
,
n

f

{R_{1,n}}f

on

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

if and only if the curve

ϕ
\phi

intersects almost every affine hyperplane. Define a measure on

M
ϕ

(

R

n

)

{M_\phi }({{\mathbf {R}}^n})

by

d
μ
=
d

μ
x

(
π
)
d
λ
(
x
)

d\mu = d{\mu _x}(\pi )d\lambda (x)

, where

d

μ
x

d{\mu _x}

is the probability measure on

M

1
,
n

{M_{1,n}}

carried by the set of lines passing through the point

x
x

and invariant under the stabilizer of

x
x

in

O
(
n
)

O(n)

and

d
λ

d\lambda

is the usual measure on

ϕ
\phi

. We show that, if

n
>
2

n > 2

and

ϕ
\phi

is unbounded, then
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper R Subscript 1 comma n Baseline f double-vertical-bar Subscript upper L Sub Superscript q Subscript left-parenthesis upper M Sub Subscript phi Subscript left-parenthesis bold upper R Sub Superscript n Subscript right-parenthesis comma d mu right-parenthesis Baseline less-than-or-equal-to upper C double-vertical-bar f double-vertical-bar Subscript upper L Sub Superscript p Subscript left-parenthesis bold upper R Sub Superscript n Subscript right-parenthesis">

‖

R

1
,
n

f
‖

L
q

(

M
ϕ

(

R

n

)
,
d
μ
)

≤
C

‖
f
‖

L
p

(

R

n

)

\left \| {R_{1,n}}f\right \|_{{L^q}({M_\phi }({{\mathbf {R}}^n}),d\mu )} \leq C\left \| f\right \| _{{L^p}({{\mathbf {R}}^n})}

if and only if

p
=
q
=
n
−
1

p = q = n - 1

and

ϕ
\phi

is line-like, that is,

λ
(
ϕ
∩
B
(
0
;
R
)
)
=
O
(
R
)

\lambda (\phi \cap B(0;R)) = O(R)

. This result gives a classification of Kirillov line complexes in terms of

L
p

{L^p}

estimates.
A simplified trace formula for Hecke operators for Γ₀(𝑁)
Authors: Ross Shepley L.
Published: June 2012

Let

N
N

and

n
n

be relatively prime positive integers, let

χ
\chi

be a Dirichlet character modulo

N
N

, and let

k
k

be a positive integer. Denote by

S
k

(
N
,
χ
)

{S_k}(N,\chi )

the space of cusp forms on

Γ
0

(
N
)

{\Gamma _0}(N)

of weight

k
k

and character

χ
\chi

, a space denoted simply

S
k

(
N
)

{S_k}(N)

when

χ
\chi

is the trivial character. Beginning with Hijikata’s formula for the trace of

T
n

{T_n}

acting on

S
k

(
N
,
χ
)

{S_k}(N,\chi )

, we develop a formula which essentially reduces the computation of this trace to looking up values in a table. From this formula we develop very simple formulas for (1) the dimension of

S
k

(
N
,
χ
)

{S_k}(N,\chi )

and (2) the trace of

T
n

{T_n}

acting on

S
k

(
N
)

{S_k}(N)

.
The structure of rings in some varieties with definable principal congruences
Authors: Simons G. E.
Published: June 2012

We study varieties of rings with identity that satisfy an identity of the form

x
y
=
y
p
(
x
,
y
)

xy = yp(x,y)

, where every term of the polynomial

p
p

has degree greater than one. These varieties are interesting because they have definable principal congruences and are residually small. Let

V

\mathcal {V}

be such a variety. The subdirectly irreducible rings in

V

\mathcal {V}

are shown to be finite local rings and are completely described. This results in structure theorems for the rings in

V

\mathcal {V}

and new examples of noncommutative rings in varieties with definable principal congruences. A standard form for the defining identity is given and is used to show that

V

\mathcal {V}

also satisfies an identity of the form

x
y
=
q
(
x
,
y
)
x

xy = q(x,y)x

. Analogous results are shown to hold for varieties satisfying

x
y
=
q
(
x
,
y
)
x

xy = q(x,y)x

.
The regular module problem. I
Authors: Berger T. R. , Hargraves B. B. , Shelton C.
Published: June 2012

In the study of induced representations the following problem arises: Let

H
=
A
G

H = AG

be a finite solvable group and

k

{\mathbf {k}}

a field with

char
⁡

k

∤

|

A

|

\operatorname {char}{\mathbf {k}}\nmid \; |A|

. Let

V
V

be an irreducible, faithful, primitive

k

[
A
G
]

{\mathbf {k}}[AG]

-module. Suppose

H
H

contains a normal extraspecial

r
r

-subgroup

R
R

with

Z
(
R
)
≤
Z
(
H
)

Z(R) \leq Z(H)

and that

A
A

acts faithfully on

R
R

. Under what conditions does

A
A

have a regular direct summand in

V
V

? In this paper we consider this question under the hypotheses that

G
=
M
R

G = MR

, where

1
≠
M

1 \ne M

is normal abelian in

A
M

AM

,

A
A

is nilpotent,
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis StartAbsoluteValue upper A EndAbsoluteValue comma StartAbsoluteValue upper M upper R EndAbsoluteValue right-parenthesis equals left-parenthesis StartAbsoluteValue upper M EndAbsoluteValue comma StartAbsoluteValue upper R EndAbsoluteValue right-parenthesis equals 1">

(

|

A

|

,

|

M
R

|

)
=
(

|

M

|

,

|

R

|

)
=
1

(|A|,|MR|) = (|M|,|R|) = 1

, and

R

/

Z
(
R
)

R/Z(R)

is a faithful, irreducible

A
M

AM

-module. We show that

A
A

has at least three regular direct summands in

V
V

unless

|

A

|

|A|

,

exp
⁡
(
M
)

\exp (M)

, and

r
r

satisfy certain very restrictive conditions.
Algorithmic procedures
Authors: Friedman Harvey , Mansfield Richard
Published: June 2012

We consider the state of elementary recursion theory when the familiar

0
,
1
,
+
,
×
,
=
,
>

0, 1, +, \times , =, >

of ordinary arithmetic are replaced by constants, functions, and relations from an arbitrary model.
The classification of complete minimal surfaces with total curvature greater than -12𝜋
Authors: López Francisco J.
Published: June 2012

We classify complete orientable minimal surfaces with finite total curvature

−
8
π

- 8\pi

.
Some integrable subalgebras of the Lie algebras of infinite-dimensional Lie groups
Authors: Leslie J.
Published: June 2012

This paper gives a proof of Lie’s second fundamental theorem in the context of infinite dimensional Lie groups; that is, we define a class of Lie subalgebras of the Lie algebra of a large class of infinite dimensional Lie groups, say

G
G

, which can be realized as the Lie algebras of Lie subgroups of

G
G

.
Nonnegative curvature on piecewise constant curvature spaces
Authors: Peszek Robert
Published: June 2012

We consider piecewise flat and piecewise spherical spaces. We prove that every piecewise flat cobordism which is a product near the boundary and has nonnegative curvature must be trivial in the metric sense. We also obtain several restrictions for piecewise

p
p

-spherical manifolds with nonnegative curvature and diameter equal to

π
p

\pi p

. We prove that such a manifold must be homeomorphic to a sphere and that it is a disjoint union of minimal paths connecting two points, which have length

π
p

\pi p

.
The Gaussian map for rational ruled surfaces
Authors: Duflot Jeanne , Miranda Rick
Published: June 2012

In this paper the Gaussian map

Φ
:

∧
2

H
0

(
C
,
K
)
→

H
0

(
C
,
3
K
)

\Phi :{ \wedge ^2}{H^0}(C,K) \to {H^0}(C,3K)

of a smooth curve

C
C

lying on a minimal rational ruled surface is computed. It is shown that the corank of

Φ
\Phi

is determined for almost all such curves by the rational surface in which it lies. Hence, except for some special cases, a curve cannot lie on two nonisomorphic minimal rational ruled surfaces.
The Mizohata structure on the sphere
Authors: Hounie Jorge
Published: June 2012

We prove that a compact surface that admits a Mizohata structure is (homeomorphic to) a sphere and that there exists exactly one such structure

L

\mathcal {L}

up to conjugation by diffeomorphisms. We also characterize the range and the kernel of the operator

δ
0

{\delta _0}

induced by

L

\mathcal {L}

, i.e., obtained from the exterior derivative on functions by passing to the quotient modulo

L

⊥

{\mathcal {L}^\bot }

.
Examples of capacity for some elliptic operators
Authors: Wu Jang-Mei
Published: June 2012

We study

L
L

-capacities for uniformly elliptic operators of nondivergence form
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L equals sigma-summation Underscript i comma j Endscripts a Subscript i j Baseline left-parenthesis x right-parenthesis StartFraction partial-differential squared Over partial-differential x Subscript i Baseline partial-differential x Subscript j Baseline EndFraction plus sigma-summation Underscript j Endscripts a Subscript j Baseline left-parenthesis x right-parenthesis StartFraction partial-differential Over partial-differential x Subscript j Baseline EndFraction semicolon">

L
=

∑

i
,
j

a

i
j

(
x
)

∂
2

∂

x
i

∂

x
j

+

∑
j

a
j

(
x
)

∂

∂

x
j

;

L = \sum \limits _{i,j} {{a_{ij}}(x)\frac {{{\partial ^2}}} {{\partial {x_i}\partial {x_j}}} + } \sum \limits _j {{a_j}(x)\frac {\partial } {{\partial {x_j}}};}

\]
and construct examples of large sets having zero

L
L

-capacity for some

L
L

, and small sets having positive

L
L

-capacity. The relations between ellipticity constants of the coefficients and the sizes of these sets are also considered.
An analogue of Siegel’s 𝜙-operator for automorphic forms for 𝐺𝐿_{𝑛}(𝑍)
Authors: Grenier Douglas
Published: June 2012

If

S

P
n

\mathcal {S}{P_n}

is the symmetric space of

n
×
n

n \times n

positive matrices,

Y
∈

S

P
n

Y \in \mathcal {S}{P_n}

can be decomposed into
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y equals Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column a m p semicolon 0 2nd Row 1st Column x 2nd Column a m p semicolon upper I EndMatrix Start 2 By 2 Matrix 1st Row 1st Column v Superscript negative 1 Baseline 2nd Column a m p semicolon 0 2nd Row 1st Column 0 2nd Column a m p semicolon v Superscript negative 1 slash left-parenthesis n minus 1 right-parenthesis Baseline upper W EndMatrix Start 2 By 2 Matrix 1st Row 1st Column 1 2nd Column a m p semicolon upper T Subscript x Baseline 2nd Row 1st Column 0 2nd Column a m p semicolon upper I EndMatrix comma">

Y
=

(

1

a
m
p
;
0

x

a
m
p
;
I

)

(

v

−
1

a
m
p
;
0

0

a
m
p
;

v

−
1

/

(
n
−
1
)

W

)

(

1

a
m
p
;

T
x

0

a
m
p
;
I

)

,

Y = \left ( {\begin {array}{*{20}{c}} 1 & 0 \\ x & I \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} {{v^{ - 1}}} & 0 \\ 0 & {{v^{ - 1/(n - 1)}}W} \\ \end {array} } \right )\left ( {\begin {array}{*{20}{c}} 1 & {{T_x}} \\ 0 & I \\ \end {array} } \right ),

\]
where

W
∈

S

P

n
−
1

W \in \mathcal {S}{P_{n - 1}}

. By letting

v
→
∞

v \to \infty

we obtain the

ϕ
\phi

-operator that attaches to every automorphic form for

G

L
n

(

Z

)

G{L_n}(\mathbb {Z})

,

f
(
Y
)

f(Y)

, an automorphic form for

G

L

n
−
1

(

Z

)

G{L_{n - 1}}(\mathbb {Z})

,

f

|

ϕ
(
W
)

f|\phi (W)

.
A spectral sequence for pseudogroups on 𝑅
Authors: Jekel Solomon M.
Published: June 2012

Consider a pseudogroup

P
P

of local homeomorphisms of

R

\mathbb {R}

satisfying the following property: given points

x
0

>
⋯
>

x
p

{x_0} > \cdots > {x_p}

and

y
0

>
⋯
>

y
p

{y_0} > \cdots > {y_p}

in

R

\mathbb {R}

, there is an element of

P
P

, with domain an interval containing

[

x
0

,

x
p

]

[{x_0},{x_p}]

, taking each

x
i

{x_i}

to

y
i

{y_i}

. The pseudogroup

P
r

{P^r}

of local

C
r

{C^r}

homeomorphisms,

0
≤
r
≤
∞

0 \leq r \leq \infty

, is of this type as is the pseudogroup

P
ω

{P^\omega }

of local real-analytic homeomorphisms. Let

Γ
\Gamma

be the topological groupoid of germs of elements of

P
P

. We construct a spectral sequence which involves the homology of a sequence of discrete groups

{

G
p

}

\{ {G_p}\}

. Consider the set

{
f
∈
P

|

f
(
i
)
=
i
,
i
=
0
,
1
,
…
,
p
}

\{ f \in P|f(i) = i,i = 0,1, \ldots ,p\}

,; define

f
∼
g

f\sim g

if

f
f

and

g
g

agree on a neighborhood of

[
0
,
p
]
⊂

R

[0,p] \subset \mathbb {R}

. The equivalence classes under composition form the group

G
p

{G_p}

. Theorem: There is a spectral sequence with

E

p
,
q

1

=

H
q

(
B

G
p

)

E_{p,q}^1 = {H_q}(B{G_p})

which converges to

H

p
+
q

(
B
Γ
)

{H_{p + q}}(B\Gamma )

. Our spectral sequence can be considered to be a version which covers the realanalytic case of some well-known theorems of J. Mather and G. Segal. The article includes some observations about how the spectral sequence applies to

B

Γ
1
ω

B\Gamma _1^\omega

. Further applications will appear separately.
Uniqueness in Cauchy problems for hyperbolic differential operators
Authors: Sogge Christopher D.
Published: June 2012

In this paper we prove a unique continuation theorem for second order strictly hyperbolic differential operators. Results also hold for higher order operators if the hyperbolic cones are strictly convex. These results are proved via certain Carleman inequalities. Unlike [6], the parametrices involved only have real phase functions, but they also have Gaussian factors. We estimate the parametrices and associated remainders using sharp

L
p

{L^p}

estimates for Fourier integral operators which are due to Brenner [1] and Seeger, Stein, and the author [5].
Motion of level sets by mean curvature. II
Authors: Evans L. C. , Spruck J.
Published: June 2012

We give a new proof of short time existence for the classical motion by mean curvature of a smooth hypersurface. Our method consists in studying a fully nonlinear uniformly parabolic equation satisfied by the signed distance function to the surface
Nilpotence and torsion in the cohomology of the Steenrod algebra
Authors: Monks Kenneth G.
Published: June 2012

In this paper we prove the existence of global nilpotence and global torsion bounds for the cohomology of any finite Hopf subalgebra of the Steenrod algebra for the prime

2
2

. An explicit formula for computing such bounds is then obtained. This is used to compute bounds for

H

∗

(

A

n

)

{H^{\ast } }({\mathcal {A}_n})

for

n
≤
6

n \leq 6

.
Subcriticality and gaugeability of the Schrödinger operator
Authors: Zhao Z.
Published: June 2012

We investigate a Schrödinger operator

−
Δ

/

2
+
V

- \Delta /2 + V

in

R
d

(
d
≥
3
)

{R^d}\;(d \geq 3)

with a potential

V
V

in the class

K
d

{K_d}

satisfying a similar Kato condition at infinity, and prove an equivalence theorem connecting various conditions on subcriticality, strong positivity and gaugeability of the operator.
Harmonic localization of algebraic 𝐾-theory spectra
Authors: Mitchell Stephen A.
Published: June 2012

The Lichtenbaum-Quillen conjectures hold for the harmonic localization of the

K
K

-theory spectrum of a nice scheme. Various consequences of this fact are explored; for example, the harmonic localization of the

K
K

-theory of the integers at a regular prime is explicitly identified.
Ramsey theory in noncommutative semigroups
Authors: Bergelson Vitaly , Hindman Neil
Published: June 2012

By utilizing ultrafilters we give a general version of the Central Sets Theorem [

6
6

, Proposition 8.21]. This enables us to derive noncommutative versions of van der Waerden’s Theorem and several of its generalizations. We also derive some standard results, including the Hales-Jewett Theorem.
On 𝑞-analogues of the Fourier and Hankel transforms
Authors: Koornwinder Tom H. , Swarttouw René F.
Published: June 2012

For H. Exton’s

q
q

-analogue of the Bessel function (going back to W. Hahn in a special case, but different from F. H. Jackson’s

q
q

-Bessel functions) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to orthogonality relations which are

q
q

-analogues of the Hankel integral transform pair. These results are implicit, in the context of quantum groups, in a paper by Vaksman and Korogodskiĭ. As a specialization we get (

q
q

-cosines and

q
q

-sines which admit

q
q

-analogues of the Fourier-cosine and Fourier-sine transforms. We also get a formula which is both an analogue of Graf’s addition formula and of the Weber-Schafheitlin discontinuous integral.
Central extensions of current algebras
Authors: Zusmanovich Paul
Published: June 2012

The second cohomology group of Lie algebras of kind

L
⊗
U

L \otimes U

with trivial coefficients is investigated, where

L
L

admits a decomposition with one-dimensional root spaces and

U
U

is an arbitrary associative commutative algebra with unit. This paper gives a unification of some recent results of C. Kassel and A. Haddi and provides a determination of central extensions of certain modular semisimple Lie algebras.
The helical transform as a connection between ergodic theory and harmonic analysis
Authors: Assani Idris , Petersen Karl
Published: June 2012

Direct proofs are given for the formal equivalence of the

L
2

{L^2}

boundedness of the maximal operators corresponding to the partial sums of Fourier series, the range of a discrete helical walk, partial Fourier coefficients, and the discrete helical transform. Strong

(
2
,
2
)

(2, 2)

for the double maximal (ergodic) helical transform is extended to actions of

R

d

{\mathbb {R}^d}

and

Z

d

{\mathbb {Z}^d}

. It is also noted that the spectral measure of a measure-preserving flow has a continuity property at

∞
\infty

, the Local Ergodic Theorem satisfies a Wiener-Wintner property, and the maximal helical transform is not weak

(
1
,
1
)

(1, 1)

.
Comparing periodic orbits of maps of the interval
Authors: Bernhardt C. , Coven E. , Misiurewicz M. , Mulvey I.
Published: June 2012

Let

π
\pi

and

θ
\theta

be cyclic permutations of finite ordered sets. We say that

π
\pi

forces

θ
\theta

if every continuous map of the interval which has a representative of

π
\pi

also has one of

θ
\theta

. We give a geometric version of Jungreis’ combinatorial algorithm for deciding in certain cases whether

π
\pi

forces

θ
\theta

.
Strong Tutte functions of matroids and graphs
Authors: Zaslavsky Thomas
Published: June 2012

A strong Tutte function of matroids is a function of finite matroids which satisfies

F
(

M
1

⊕

M
2

)
=
F
(

M
1

)
F
(

M
2

)

F({M_1} \oplus {M_2}) = F({M_1})F({M_2})

and

F
(
M
)
=

a
e

F
(
M
∖
e
)
+

b
e

F
(
M

/

e
)

F(M) = {a_e}F(M\backslash e) + {b_e}F(M/e)

for

e
e

not a loop or coloop of

M
M

, where

a
e

{a_e}

,

b
e

{b_e}

are scalar parameters depending only on

e
e

. We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski’s classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial. The classification remains valid if the domain is any minor-closed class of matroids containing all three-point matroids. Similar classifications hold for strong Tutte functions of colored matroids, where the parameters depend on the color of

e
e

, and for strong Tutte functions of graphs and edge-colored graphs whose values do not depend on the attachments of loops. The latter classification implies new characterizations of Kauffman’s bracket polynomials of signed graphs and link diagrams.
Differentials of complex interpolation processes for Köthe function spaces
Authors: Kalton N. J.
Published: June 2012

We continue the study of centralizers on Köthe function spaces and the commutator estimates they generate (see [29]). Our main result is that if

X
X

is a super-reflexive Köthe function space then for every real centralizer

Ω
\Omega

on

X
X

there is a complex interpolation scale of Köthe function spaces through

X
X

inducing

Ω
\Omega

as a derivative, up to equivalence and a scalar multiple. Thus, in a loose sense, all real centralizers can be identified with derivatives of complex interpolation processes. We apply our ideas in an appendix to show, for example, that there is a twisted sum of two Hilbert spaces which fails to be a

(

UMD

)

({\text {UMD}})

-space.
Characteristic cycles of holomorphic discrete series
Authors: Chang Jen-Tseh
Published: June 2012

The characteristic cycles of standard modules arising from closed

K
K

-orbits in a flag variety are studied and those of holomorphic discrete series are determined explicitly. Also the asymptotic expansion of the distribution characters of holomorphic discrete series are computed; the result verifies a conjecture of D. Vogan in this special case.
Uniqueness of radial solutions of semilinear elliptic equations
Authors: Kwong Man Kam , Li Yi
Published: June 2012

E. Yanagida recently proved that the classical Matukuma equation with a given exponent has only one finite mass solution. We show how similar ideas can be exploited to obtain uniqueness results for other classes of equations as well as Matukuma equations with more general coefficients. One particular example covered is

Δ
u
+

u
p

±
u
=
0

\Delta u + {u^p} \pm u = 0

, with

p
>
1

p > 1

. The key ingredients of the method are energy functions and suitable transformations. We also study general boundary conditions, using an extension of a recent result by Bandle and Kwong. Yanagida’s proof does not extend to solutions of Matukuma’s equation satisfying other boundary conditions. We treat these with a completely different method of Kwong and Zhang.
Cohomological aspects of hypergraphs
Authors: Chung F. R. K. , Graham R. L.
Published: June 2012

By a

k
k

-graph we will mean a collection of

k
k

-element subsets of some fixed set

V
V

. A

k
k

-graph can be regarded as a

(
k
−
1
)

(k - 1)

-chain on

2
V

{2^V}

, the simplicial complex of all subsets of

V
V

, over the coefficient group

Z

/

2

\mathbb {Z}/2

, the additive group of integers modulo

2
2

. The induced group structure on the

(
k
−
1
)

(k - 1)

-chains leads to natural definitions of the coboundary

δ
\delta

of a chain, the cochain complex of

C
=
{

C
k

,
δ
}

C = \{ {C^k},\delta \}

and the usual cohomology groups

H
k

(
C
;

Z

/

2
)

{H^k}(C;\mathbb {Z}/2)

. In particular, it is possible to construct what could be called "higher-order" coboundary operators

δ

(
i
)

{\delta ^{(i)}}

, where

δ

(
i
)

{\delta ^{(i)}}

increases dimension by

i
i

(rather than just

1
1

). In this paper we will develop various properties of these

δ

(
i
)

{\delta ^{(i)}}

, and in particular, compute the corresponding cohomology groups for

2
V

{2^V}

over

Z

/

2

\mathbb {Z}/2

. It turns out that these groups depend in a rather subtle way on the arithmetic properties of

i
i

.
Generalized Toda brackets and equivariant Moore spectra
Authors: Costenoble Steven R. , Waner Stefan
Published: June 2012

In this paper we develop a general theory of obstructions to the existence of equivariant Moore spectra. The obstructions we obtain coincide with higher order Toda brackets as defined by Spanier. We then apply the theory to show the existence of equivariant Moore spectra in various special cases.
Automorphisms of the lattice of recursively enumerable sets: promptly simple sets
Authors: Cholak P. , Downey R. , Stob M.
Published: June 2012

We show that for every coinfinite r.e. set

A
A

there is a complete r.e. set

B
B

such that

L

∗

(
A
)

≈

eff

L

∗

(
B
)

{\mathcal {L}^{\ast } }(A){ \approx _{{\text {eff}}}}{\mathcal {L}^{\ast } }(B)

and that every promptly simple set is automorphic (in

E

∗

{\mathcal {E}^{\ast } }

) to a complete set.
On automorphisms of Markov chains
Authors: Krieger Wolfgang , Marcus Brian , Tuncel Selim
Published: June 2012

We prove several theorems about automorphisms of Markov chains, using the weight-per-symbol polytope.
Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter
Authors: Mantlik Frank
Published: June 2012

Let
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P left-parenthesis lamda comma upper D right-parenthesis equals sigma-summation Underscript StartAbsoluteValue alpha EndAbsoluteValue less-than-or-equal-to m Endscripts a Subscript alpha Baseline left-parenthesis lamda right-parenthesis upper D Superscript alpha">

P
(
λ
,
D
)
=

∑

|

α

|

≤
m

a
α

(
λ
)

D
α

P(\lambda ,D) = \sum \nolimits _{|\alpha | \leq m} {{a_\alpha }(\lambda ){D^\alpha }}

be a differential operator with constant coefficients

a
α

{a_\alpha }

depending analytically on a parameter

λ
\lambda

. Assume that each

P
(
λ
,
D
)

P(\lambda ,D)

is hypoelliptic and that the strength of

P
(
λ
,
D
)

P(\lambda ,D)

is independent of

λ
\lambda

. Under this condition we show that there exists a regular fundamental solution of

P
(
λ
,
D
)

P(\lambda ,D)

which also depends analytically on

λ
\lambda

.
Hypoellipticity on Cauchy-Riemann manifolds
Authors: Petersen Johannes A.
Published: June 2012

Using a recent homotopy formula by Trèves, it is shown that the existence of

(
q
+
1
)

(q + 1)

-dimensional holomorphic supporting manifolds is a sufficient condition for the hypoellipticity on level

q
q

and

n
−
q

n - q

of a tangential Cauchy-Riemann complex of

CR

{\text {CR}}

-dimension

n
n

. In the hypersurface case, this result is given microlocally.
𝑣₁-periodic homotopy groups of exceptional Lie groups: torsion-free cases
Authors: Bendersky Martin , Davis Donald M. , Mimura Mamoru
Published: June 2012

The

v
1

{v_1}

-periodic homotopy groups

v
1

−
1

π

∗

(
X
;
p
)

v_1^{ - 1}{\pi _ {\ast } }(X;p)

are computed explicitly for all pairs

(
X
,
p
)

(X,p)

, where

X
X

is an exceptional Lie group whose integral homology has no

p
p

-torsion. This yields new lower bounds for

p
p

-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof.