bluuebunny/crossref_metadata_2025_split huggingface dataset:
Direct computation of the spectral function
Authors: Boumenir Amin 
Published: June 2010

We would like to find an explicit formula for the spectral function of the following Sturm-Liouville problem:
\[

{

L
f
≡
−

d
2

d

x
2

f
(
x
)
+
q
(
x
)
f
(
x
)
,

x
≥
0
,

f
′

(
0
)
−
m
f
(
0
)
=
0.

\left \{ {\begin {array}{*{20}{c}} {Lf \equiv - \frac {{{d^2}}}{{d{x^2}}}f(x) + q(x)f(x),\quad x \geq 0,} \hfill \\ {f’(0) - mf(0) = 0.} \hfill \\ \end {array} } \right .

\]
A simple operational calculus argument will help us obtain an explicit formula for the transmutation kernel. The expression of the spectral function is then obtained through the nonlinear integral equation found in the Gelfand-Levitan theory.
A collection of sequence spaces
Authors: Calder J. R. , Hill J. B. 
Published: June 2010

This paper concerns a collection of sequence spaces we shall refer to as

d
α

{d_\alpha }

spaces. Suppose

α
=
(

α
1

,

α
2

,
…
)

\alpha = ({\alpha _1},{\alpha _2}, \ldots )

is a bounded number sequence and

α
i

≠
0

{\alpha _i} \ne 0

for some

i
i

. Suppose

P

\mathcal {P}

is the collection of permutations on the positive integers. Then

d
α

{d_\alpha }

denotes the set to which the number sequence

x
=
(

x
1

,

x
2

,
…
)

x = ({x_1},{x_2}, \ldots )

belongs if and only if there exists a number

k
>
0

k > 0

such that
\[

h
α

(
x
)
=

lub

p
∈

P

⁡

∑

i
=
1

∞

|

x

F
(
i
)

α
i

|

>
k
.

h_\alpha (x) = \operatorname {lub}_{p \in \mathcal {P}} \sum \limits _{i = 1}^\infty |x_{F(i)} \alpha _i| > k.

\]

h
α

h_\alpha

is a norm on

d
α

d_\alpha

and

(

d
α

,

h
α

)

(d_\alpha , h_\alpha )

is complete. We classify the

d
α

{d_\alpha }

spaces and compare them with

l
1

{l_1}

and

m
m

. Some of the

d
α

{d_\alpha }

spaces are shown to have a semishrinking basis that is not shrinking. Further investigation of the bases in these spaces yields theorems concerning the conjugate space properties of

d
α

{d_\alpha }

. We characterize the sequences

β
\beta

such that, given

α
,

d
β

,
=

d
α

\alpha ,{d_\beta }, = {d_\alpha }

. A class of manifolds in the first conjugate space of

d
α

{d_\alpha }

is examined. We establish some properties of the collection of points in the first conjugate space of a normed linear space

S
S

that attain their maximum on the unit ball in

S
S

. The effect of renorming

c
0

{c_0}

and

l
1

{l_1}

with

h
α

{h_\alpha }

and related norms is studied in terms of the change induced on this collection of functionals.
Generic Fréchet-differentiability and perturbed optimization problems in Banach spaces
Authors: Ekeland Ivar , Lebourg Gérard 
Published: June 2010

We define a function F on a Banach space V to be locally

ε
\varepsilon

-supported by

u
∗

∈

V
∗

{u^\ast } \in {V^\ast }

at

u
∈
V

u \in V

if there exists an

η
>
0

\eta > 0

such that

‖

v
−
u

‖

⩽
η
⇒
F
(
v
)
⩾
F
(
u
)
+
⟨

u
∗

,
v
−
u
⟩
−
ε

‖

v
−
u

‖

\left \| {v - u} \right \| \leqslant \eta \Rightarrow F(v) \geqslant F(u) + \langle {u^\ast },v - u\rangle - \varepsilon \left \| {v - u} \right \|

. We prove that if the Banach space V admits a nonnegative Fréchet-differentiable function with bounded nonempty support, then, for any

>
0

> 0

and every lower semicontinuous function F, there is a dense set of points

u
∈
V

u \in V

at which F is locally

ε
\varepsilon

-supported. The applications are twofold. First, to the study of functions defined as pointwise infima; we prove for instance that every concave continuous function defined on a Banach space with Fréchet-differentiable norm is Fréchet-differentiable generically (i.e. on a countable intersection of open dense subsets). Then, to the study of optimization problems depending on a parameter

u
∈
V

u \in V

; we give general conditions, mainly in the framework of uniformly convex Banach spaces with uniformly convex dual, under which such problems generically have a single optimal solution, depending continuously on the parameter and satisfying a first-order necessary condition.
Derivatives of entire functions and a question of Pólya. II
Authors: Hellerstein Simon , Williamson Jack 
Published: June 2010

It is shown that if f is an entire function of infinite order, which is real on the real axis and has, along with

f
′

f’

, only real zeros, then

f
f

has nonreal zeros (in fact, infinitely many). The finite order case was treated by the authors in a preceding paper. The combined results show that the only real entire functions f for which

f
,

f
′

f,f’

, and

f
f

have only real zeros are those in the Laguerre-Pólya class, i.e.
\[

f
(
z
)
=

z
m

exp
⁡
{
−
a

z
2

+
b
z
+
c
}

∏
n

(

1
−

z

z
n

)

e

z

/

z
n

,

f(z) = {z^m}\exp \{ - a{z^2} + bz + c\} \prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}},

\]

a
⩾
0
,
b
,
c

a \geqslant 0,b,c

and the

z
n

{z_n}

real, and

Σ

z
n

−
2

>
∞

\Sigma z_n^{ - 2} > \infty

. This gives a strong affirmative version of an old conjecture of Pólya.
Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials
Authors: Hazewinkel Michiel 
Published: June 2010

For any ring

R
R

let

Λ
(
R
)

\Lambda (R)

denote the multiplicative group of power series of the form

1
+

a
1

t
+
⋯

1 + {a_1}t + \cdots

with coefficients in

R
R

. The Artin-Hasse exponential mappings are homomorphisms

W

p
,
∞

(
k
)
→
Λ
(

W

p
,
∞

(
k
)
)

W_{p, \infty } (k) \to \Lambda ({W_{p, \infty }}(k))

, which satisfy certain additional properties. Somewhat reformulated, the Artin-Hasse exponentials turn out to be special cases of a functorial ring homomorphism

E
:

W

p
,
∞

(
−
)
→

W

p
,
∞

(

W

p
,
∞

(
−
)
)

E: {W_{p, \infty }}( - ) \to {W_{p,\infty }}({W_{p,\infty }}( - ))

, where

W

p
,
∞

{W_{p,\infty }}

is the functor of infinite-length Witt vectors associated to the prime

p
p

. In this paper we present ramified versions of both

W

p
,
∞

(
−
)

{W_{p,\infty }}( - )

and

E
E

, with

W

p
,
∞

(
−
)

{W_{p,\infty }}( - )

replaced by a functor

W

q
,
∞

F

(
−
)

W_{q,\infty }^F( - )

, which is essentially the functor of

q
q

-typical curves in a (twisted) Lubin-Tate formal group law over

A
A

, where

A
A

is a discrete valuation ring that admits a Frobenius-like endomorphism

σ
\sigma

(we require

σ
(
a
)
≡

a
q

mod

m

\sigma (a) \equiv {a^q} \bmod \mathfrak {m}

for all

a
∈
A

a \in A

, where

m

\mathfrak {m}

is the maximal idea of

A
A

). These ramified-Witt-vector functors

W

q
,
∞

F

(
−
)

W_{q,\infty }^F( - )

do indeed have the property that, if

k
=
A

/

m

k = A/\mathfrak {m}

is perfect,

A
A

is complete, and

l

/

k

l/k

is a finite extension of

k
k

, then

W

q
,
∞

F

(
l
)

W_{q,\infty }^F(l)

is the ring of integers of the unique unramified extension

L

/

K

L/K

covering

l

/

k

l/k

.
On the topology of simply connected algebraic surfaces
Authors: Mandelbaum Richard , Moishezon Boris 
Published: June 2010

Suppose x is a smooth simply-connected compact 4-manifold. Let

p

=

C

P
2

p\, = \,{\textbf {C}}{P^2}

and

Q

=

−

C

P
2

Q\, = \, - {\textbf {C}}{P^2}

be the complex projective plane with orientation opposite to the usual. We shall say that X is completely decomposable if there exist integers a, b such that X is diffeomorphic to

a
P

\#

b
Q

aP\,{\text {\# }}\,bQ

. By a result of Wall [W1] there always exists an integer k such that

X

#

(
k

+

1
)
P

#
k
Q

X\,\# \,(k\, + \,1)P\,\# kQ

is completely decomposable. If

X

#

P

X\,\# \,P

is completely decomposable we shall say that X is almost completely decomposable. In [MM] we demonstrated that any nonsingular hypersurface of

C

P
3

{\textbf {C}}{P^3}

is almost completely decomposable. In this paper we generalize this result in two directions as follows: Theorem 3.5. Suppose W is a simply-connected nonsingular complex projective 3-fold. Then there exists an integer

m
0

⩾

1

{m_0}\, \geqslant \,1

such that any hypersurface section

V
m

{V_m}

of W of degree

m

⩾

m
0

m\, \geqslant \,{m_0}

which is nonsingular will be almost completely decomposable. Theorem 5.3. Let V be a nonsingular complex algebraic surface which is a complete intersection. Then V is almost completely decomposable.
Countable ultrahomogeneous undirected graphs
Authors: Lachlan A. H. , Woodrow Robert E. 
Published: June 2010

Let

G

=

⟨

V
G

,

E
G

⟩

G\, = \,\left \langle {{V_G},\,{E_G}} \right \rangle

be an undirected graph. The complementary graph

G
~

\tilde G

is

⟨

V
G

,

E

G
~

⟩

\left \langle {{V_G},\,{E_{\tilde G}}} \right \rangle

where

(

V
1

,

V
2

)

∈

E

G
~

({V_1},\,{V_2})\, \in \,{E_{\tilde G}}

iff

V
1

≠

V
2

{V_1}\, \ne \,{V_2}

and

(

V
1

,

V
2

)

∉

E
G

({V_1},\,{V_2})\, \notin \,{E_G}

. Let

K
(
n
)

K(n)

be the complete undirected graph on n vertices and let E be the graph [ill] i.e.

⟨

{
a
,

b
,

c
}
,

{
(
b
,

c
)
,

(
c
,

b
)
}

⟩

\left \langle {\{ a,\,b,\,c\} ,\,\{ (b,\,c),\,(c,\,b)\} } \right \rangle

. G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to an automorphism of G. Let

D

=

{
K
(
n
)
:

n

∈

ω
}

∪

{
E
,

E
~

}

∪

{

K
~

(
n
)
:

n

∈

ω
}

\mathcal {D}\, = \,\{ K(n):\,n\, \in \,\omega \} \, \cup \,\{ E,\,\tilde E\} \, \cup \,\{ \tilde K(n):\,n\, \in \,\omega \}

. Theorem: Let

G
1

{G_1}

,

G
2

{G_2}

be two countable (infinite) ultrahomogeneous graphs such that for each

H

∈

D

H

H\, \in \,\mathcal {D}\,H

can be embedded in

G
1

{G_1}

, just in case it can be embedded in

G
2

{G_2}

. Then

G
1

≅

G
2

{G_1}\, \cong \,{G_2}

. Corollary: There are a countable number of countable ultrahomogeneous (undirected) graphs.
Weak subordination and stable classes of meromorphic functions
Authors: Stephenson Kenneth 
Published: June 2010

This paper introduces the notion of weak subordination: If

F
F

and

G
G

are meromorphic in the unit disc

U

\mathcal {U}

, then

F
F

is weakly subordinate to

G
G

, written

F
>
G

F > G

, provided there exist analytic functions

ϕ
\phi

and

ω
:

U

→

U

\omega :\mathcal {U} \to \mathcal {U}

, with

ϕ
\phi

an inner function, so that

F
∘
ϕ
=
G
∘
ω

F \circ \phi = G \circ \omega

. A class

X

\mathcal {X}

of meromorphic functions is termed stable if

F

≺

w

G

F \stackrel {w}{\prec } G

and

G
∈

X

⇒
F
∈

X

G \in \mathcal {X} \Rightarrow F \in \mathcal {X}

. The motivation is recent work of Burkholder which relates the growth of a function with its range and boundary values. Assume

F
F

and

G
G

are meromorphic and

G
G

has nontangential limits, a.e. Assume further that

F
(

U

)
∩
G
(

U

)
≠
∅

F(\mathcal {U}) \cap G(\mathcal {U}) \ne \emptyset

and

G
(

e

i
θ

)
∉
F
(

U

)

G({e^{i\theta }}) \notin F(\mathcal {U})

, a.e. This is denoted by

F
>
G

F > G

. Burkholder proved for several classes

X

\mathcal {X}

that

(

(
∗
)

)

F
>
G

and

G
∈

X

⇒
F
∈

X

.

\begin{equation}\tag {$(\ast )$}F > G \qquad {\text {and}}\quad G \in \mathcal {X} \Rightarrow F \in \mathcal {X}.\end{equation}

The main result of this paper is the Theorem:

F
>
G
⇒
F

≺
w

G

F > G \Rightarrow F{ \prec ^w}G

. In particular, implication (*) holds for all stable classes

X

\mathcal {X}

. The paper goes on to study various stable classes, which include BMOA,

H
p

{H^p}

,

0
>
p
⩽
∞

0 > p \leqslant \infty

,

N

∗

{N_{\ast }}

, the space of functions of bounded characteristic, and the

M
Φ

{M^\Phi }

spaces introduced by Burkholder. VMOA and the Bloch functions are examples of classes which are not stable.
A maximal function characterization of 𝐻^{𝑝} on the space of homogeneous type
Authors: Uchiyama Akihito 
Published: June 2010

Let

ψ
0

(
x
)

∈

S
(

R
n

)

{\psi _0}(x)\, \in \,{\mathcal {S(}}{R^n}{\text {)}}

and let

∫

R
n

ψ
0

(
y
)

d
y

≠

0

\int _{{R^n}} {{\psi _0}(y)\,dy\, \ne \,0}

.For

f

∈

S

′

R
n

f\, \in \,{\mathcal {S}’}{R^n}

,

x

∈

R
n

x\, \in \,{R^n}

and

M

⩾

0

M\, \geqslant \,0

, let
\[

f
+

(
x
)

=

sup

t

>

0

|

f

∗

ψ

0
t

(
x
)

|

{f^ + }(x)\, = \,\sup \limits _{t\, > \,0} \,\left | {f\,{\ast }\,{\psi _{0t}}(x)} \right |

\]
and let

f

∗

M

(
x
)

=

sup
{

|

f

∗

ψ
t

(
x
)

|

:

t

>

0

{f^{{\ast }M}}(x)\, = \,\sup \{ \left | {f\,{\ast }\,{\psi _t}(x)} \right |:\,t\, > \,0

,

ψ
(
y
)

∈

S
(

R
n

)

\psi (y)\, \in \,{\mathcal {S(}}{R^n})

,

supp

ψ

⊂

{
y

∈

R
n

:

|
y
|

>

1
}

\operatorname {supp} \,\psi \, \subset \,\{ y\, \in \,{R^n}:\,\left | y \right |\, > \,1\}

,

‖

D
α

ψ

‖

L
∞

⩽

1

{\left \| {{D^\alpha }\psi } \right \|_{{L^\infty }}}\, \leqslant \,1

for any multi-index

α

=

(

α
1

,

…
,

α
n

)

\alpha \, = \,({\alpha _1},\, \ldots ,\,{\alpha _n})

such that

Σ

i
=
1

n

α
i

⩽

M
}

\Sigma _{i = 1}^n\,{\alpha _i}\, \leqslant \,M\}

where

ψ
t

(
y
)

=

t

−
n

ψ
(
y

/

t
)

{\psi _t}(y)\, = \,{t^{ - n}}\psi (y/t)

. Fefferman-Stein [11] showed Theorem A. Let

p

>

0

p\, > \,0

. Then there exists

M
(
p
,

n
)

M(p,\,n)

, depending only on p and n, such that if

M

⩾

M
(
p
,

n
)

M\, \geqslant \,M(p,\,n)

, then
\[

c

‖

f
+

‖

L
p

⩽

‖

f

∗

M

‖

L
p

⩽

C

‖

f
+

‖

L
p

c\left \| {{f^ + }} \right \|{L^p}\, \leqslant \,\left \| {{f^{{\ast }M}}} \right \|{L^p}\, \leqslant \,{\textbf {C}}\left \| {{f^ + }} \right \|{L^p}

\]
for any

f

∈

S

′

(

R
n

)

f\, \in \,{\mathcal {S}’}({R^n})

, where c and C are positive constants depending only on

ψ
0

{\psi _0}

, p, M and n. We investigate this on the space of homogeneous type with certain assumptions.
Integral geometric properties of capacities
Authors: Mattila Pertti 
Published: June 2010

Let

m
m

and

n
n

be positive integers,

0
>
m
>
n

0 > m > n

, and

C
K

{C_K}

and

C
H

{C_H}

the usual potential-theoretic capacities on

R
n

{R^n}

corresponding to lower semicontinuous kernels

K
K

and

H
H

on

R
n

×

R
n

{R^n} \times {R^n}

with

H
(
x
,
y
)
=
K
(
x
,
y
)

|

x
−
y

|

n
−
m

⩾
1

H(x,y) = K(x,y){\left | {x - y} \right |^{n - m}} \geqslant 1

for

|

x
−
y

|

⩽
1

\left | {x - y} \right | \leqslant 1

. We consider relations between the capacities

C
K

(
E
)

{C_K}(E)

and

C
H

(
E
∩
A
)

{C_H}(E \cap A)

when

E
⊂

R
n

E \subset {R^n}

and

A
A

varies over the

m
m

-dimensional affine subspaces of

R
n

{R^n}

. For example, we prove that if

E
E

is compact,

C
K

(
E
)
⩽
c
∫

C
H

(
E
∩
A
)
d

λ

n
,
m

A

{C_K}(E) \leqslant c\smallint {C_H}(E \cap A)d{\lambda _{n,m}}A

where

λ

n
,
m

{\lambda _{n,m}}

is a rigidly invariant measure and

c
c

is a positive constant depending only on

n
n

and

m
m

.
Disproof of a coefficient conjecture for meromorphic univalent functions
Authors: Tsao Anna 
Published: June 2010

Let

Σ
\Sigma

denote the class of functions

g
(
z
)
=
z
+

b
0

+

b
1

z

−
1

+
⋯

g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots

analytic and univalent in

|

z

|

>
1

|z|> 1

except for a simple pole at

∞
\infty

. A well-known conjecture asserts that

|

b
n

|

≤
2

/

(
n
+
1
)

(
n
=
1
,
2
,
…
)

|{b_n}|\, \leq 2/(n + 1)\qquad (n = 1,2, \ldots )

with equality for

g
(
z
)
=

(
1
+

z

n
+
1

)

2

/

(
n
+
1
)

/

z
=
z
+
2

z

−
n

/

(
n
+
1
)
+
⋯

g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots

. Although the conjecture is true for

n
=
1
,
2

n=1,2

and certain subclasses of the class

Σ
\Sigma

, the general conjecture is known to be false for all odd

n
≥
3

n\ge 3

and

n
=
4

n=4

. In

§
2

\S 2

, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in

§
3

\S 3

to construct explicit counterexamples to the conjecture for all

n
>
4

n > 4

. In fact, the conjectured extremal function does not even provide a local maximum for

Re

{

b
n

}

{\text {Re}}\{ {b_n}\}

,

n
>
4

n > 4

.
Some canonical cohomology classes on groups of volume preserving diffeomorphisms
Authors: McDuff Dusa 
Published: June 2010

We discuss some canonical cohomology classes on the space

B
¯

D

i
f

f

ω
0

c

M

\bar B\mathcal {D}iff_{\omega 0}^cM

, where

D

i
f

f

ω
0

c

M

\mathcal {D}iff_{\omega 0}^cM

is the identity component of the group of compactly supported diffeomorphisms of the manifold

M
M

which preserve the volume form

ω
\omega

. We first look at some classes

c
k

(
M
)
,
1
⩽
k
⩽
n
=

dim

M

{c_k}(M),1 \leqslant k \leqslant n = {\text {dim}}\,M

, which are defined for all

M
M

, and show that the top class

c
n

(
M
)
∈

H
n

(

B
¯

D

i
f

f

ω
0

c

M
;

R

)

{c_n}(M) \in \,{H^n}(\bar B\mathcal {D}iff_{\omega 0}^cM;{\mathbf {R}})

is nonzero for

M
=

S
n

,
n

M = {S^n},n

odd, and is zero for

M
=

S
n

,
n

M = {S^n},n

even. When

H
c
i

(
M
;

R

) = 0

H_c^i(M;{\mathbf {R}}{\text {) = 0}}

for

0
⩽
i
>
n

0 \leqslant i > n

, the classes

c
k

(
M
)

{c_k}(M)

all vanish and a secondary class

s
(
M
)
∈

H

n
−
1

(

B
¯

D

i
f

f

ω
0

c

M
;

R

)

s(M) \in \,{H^{n - 1}}(\bar B\mathcal {D}iff_{\omega 0}^cM; {\mathbf {R}})

may be defined. This is trivially zero when

n
n

is odd, and is twice the Calabi invariant for symplectic manifolds when

n
=
2

n = 2

. We prove that

s
(

R

n

)
≠
0

s({{\mathbf {R}}^n}) \ne 0

when

n
n

is even by showing that it is one of a set of nonzero classes which were defined by Hurder in [7].
Generalized intersection multiplicities of modules
Authors: Dutta Sankar P. 
Published: June 2010

In this paper we study intersection multiplicities of modules as defined by Serre and prove that over regular local rings of

dim
⩽
5

\dim \leqslant 5

, given two modules

M
,
N

M,N

with

l
(
M

⊗

R

N
)
>
∞

l(M\otimes _{R}N) > \infty

and

dim

M
+
dim

N
>
dim

R
,
χ
(
M
,
N
)
=

∑

i
=
0

dim

R

(
−
1

)
i

l
(

Tor
i
R

⁡
(
M
,
N
)
)
=
0

\dim \;M + \dim \;N > \dim \;R,\chi (M,N) = \sum \nolimits _{i = 0}^{\dim \; R}( - 1)^i l(\operatorname {Tor}_i^R(M,N)) = 0

. We also study multiplicity in a more general set up. Finally we extend Serre’s result from pairs of modules to pairs of finite free complexes whose homologies are killed by

I
n

,

J
n

{I^n},{J^n}

, respectively, for some

n
>
0

n > 0

, with

dim

R

/

I
+
dim

R

/

J
>
dim

R

\dim \,R/I + \dim \,R/J > \dim \,R

.
Poles of a two-variable 𝑃-adic complex power
Authors: Strauss Leon 
Published: June 2010

For almost all

P
P

-adic completions of an algebraic number field, if

s
∈

C

s \in {\mathbf {C}}

is a pole of

f
s

=

∫

∫

|

f
(
x
,
y
)

|

s

|

d
x

|

K
p

|

d
y

|

K
p

{f^s} = \int _{}^{} {\int _{}^{} {|f(x,y){|^s}|dx{|_{{K_p}}}|dy{|_{{K_p}}}} }

, where

f
f

is a polynomial whose only singular point is the origin,

f
(
0
,
0
)
=
0

f(0,0) = 0

, and

f
f

is irreducible in

K
¯

[
[
x
,
y
]
]

\overline K [[x,y]]

, then

Re
⁡
(
s
)

\operatorname {Re} (s)

is

−
1

- 1

or one of an explicitly given set of rational numbers, whose cardinality is the number of characteristic exponents of

f
=
0

f = 0

.
A strong type (2,2) estimate for a maximal operator associated to the Schrödinger equation
Authors: Kenig Carlos E. , Ruiz Alberto 
Published: June 2010

Let

T

∗

f
(
x
)
=

sup

t
>
0

|

T
t

f
(
x
)

|

{T^{\ast } }f(x) = \sup _{t > 0}|{T_t}f(x)|

, where

(

T
t

f

)

^

(
ξ
)
=

e

i
t

|

ξ

|

2

f
^

(
ξ
)

/

|

ξ

|

1

/

4

({T_t}f)^{\hat {}}(\xi ) = {e^{it|\xi |^2}}\hat f(\xi )/|\xi {|^{1/4}}

. We show that, given any finite interval

I
I

,

∫
I

|

T

∗

f

|

2

d
x
⩽

C
I

∫

R

|

f
(
x
)

|

2

d
x

\int _I {|{T^{\ast } }f{|^2}\;dx \leqslant {C_I}\int _{\mathbf {R}} {|f(x){|^2}\;dx} }

, and that the above inequality is false with

2
2

replaced by any

p
>
2

p > 2

. This maximal operator is related to solutions of the Schrödinger equation.
The 𝑆¹-transfer in surgery theory
Authors: Munkholm H. J. , Pedersen E. K. 
Published: June 2010

Let

S
1

→
X
→
Y

{S^1} \to X \to Y

be an

S
1

{S^1}

-bundle of Poincaré spaces. If

f
:
N
→
Y

f:N \to Y

is a surgery problem then so is the pullback

f
^

:
M
→
X

\hat f:M \to X

. We define algebraically a homomorphism

φ
!

:

L
n

(

Z

π
1

(
Y
)
)
→

L

n
+
1

(

Z

π
1

(
X
)
)

{\varphi ^!}:{L_n}({\mathbf {Z}}{\pi _1}(Y)) \to {L_{n + 1}}({\mathbf {Z}}{\pi _1}(X))

and prove that it maps the surgery obstruction for

f
f

to the one for

f
^

\hat f

.
James maps and 𝐸_{𝑛} ring spaces
Authors: Cohen F. R. , May J. P. , Taylor L. R. 
Published: June 2010

We parametrize by operad actions the multiplicative analysis of the total James map given by Caruso and ourselves. The target of the total James map
\[

j
=
∑

j
q

:
C
(

R
n

,
X
)
→

∏

q
⩾
0

Q

D
q

(

R
n

,
X
)

j = \sum {{j_q}} :C({R^n},X) \to \prod \limits _{q \geqslant 0} {Q{D_q}({R^n},X)}

\]
is an

E
n

{E_n}

ring space and

j
j

is a

C

n

{\mathcal {C}_n}

-map, where

C

n

{\mathcal {C}_n}

is the little

n
n

-cubes operad. This implies that

j
j

has an

n
n

-fold delooping with domain

Σ
n

X

{\Sigma ^n}X

. It also implies an algorithm for the calculation of

j

∗

{j_{\ast }}

and thus of each

(

j
q

)

∗

{({j_q})_{\ast }}

on

mod

p

\bmod \, p

homology. When

n
=
∞

n = \infty

and

p
=
2

p = 2

, this algorithm is the essential starting point for Kuhn’s proof of the Whitehead conjecture.
Orthogonal polynomials on the sphere with octahedral symmetry
Authors: Dunkl Charles F. 
Published: June 2010

For any finite reflection group

G
G

acting on

R

N

{{\mathbf {R}}^N}

there is a family of

G
G

-invariant measures (

(

h
2

d
ω

({h^2}d\omega

, where

h
h

is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for

G
G

) on the unit sphere and an associated partial differential operator (

L
h

f
:=
Δ
(
f
h
)
−
f
Δ
h

{L_h}f: = \Delta (fh) - f\Delta h

;

Δ
\Delta

is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to

h
2

d
ω

{h^2}d\omega

) decomposition of homogeneous polynomials, that is, if

p
p

is of degree

n
n

then
\[

p
(
x
)
=

∑

j
=
0

[
n

/

2
]

|

x

|

2
j

p

n
−
2
j

(
x
)
,

p(x) = \sum \limits _{j = 0}^{[n/2]} {|x{|^{2j}}{p_{n - 2j}}(x),}

\]
where

L
h

p
i

=
0

{L_h}{p_i} = 0

and

deg

p
i

=
i

{\operatorname {deg}}{p_i} = i

for each

i
i

, but with the restriction that

p
p

and

p
i

{p_i}

must all be

G
G

-invariant. The main topic is the hyperoctahedral group with
\[

h
(
x
)
=

(

x
1

x
2

⋯

x
N

)
α

(

∏

i
>
j

(

x
i
2

−

x
j
2

)

)

β

.

h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left ( {\prod \limits _{i > j} {(x_i^2 - x_j^2)} } \right )^\beta }.

\]
The special case

N
=
2

N = 2

leads to Jacobi polynomials. A detailed study of the case

N
=
3

N = 3

is made; an important result is the construction of a third-order differential operator that maps polynomials associated to

h
h

with indices

(
α
,
β
)

(\alpha ,\beta )

to those associated with

(
α
+
2
,
β
+
1
)

(\alpha + 2,\beta + 1)

.