bluuebunny/crossref_metadata_2025_split huggingface dataset:
Infinite nodal noncommutative Jordan algebras; differentiably simple algebras
Authors: Scribner D. R. 
Published: July 2010

The first result is that any differentiably simple algebra of the form

A
=
F
1
+
R

A = F1 + R

, for R a proper ideal, 1 the identity element, and F the base field, must be a subalgebra of a (commutative associative) power series algebra over F, and is truncated if the characteristic is not zero. Moreover the algebra A contains the polynomial subalgebra generated by the indeterminates and identity of the power series algebra. This is used to prove that if A is any simple flexible algebra of the form

A
=
F
1
+
R

A = F1 + R

, R an ideal of

A
+

{A^ + }

, then

A
+

{A^ + }

is a subalgebra of a power series algebra and multiplication in A is determined by certain elements

c

i
j

{c_{ij}}

in A as in
\[

f
g
=
f
⋅
g
+

1
2

∑

∂
f

∂

x
i

⋅

∂
g

∂

x
j

⋅

c

i
j

,

fg = f \cdot g + \frac {1}{2}\sum {\frac {{\partial f}}{{\partial {x_i}}} \cdot \frac {{\partial g}}{{\partial {x_j}}} \cdot {c_{ij}},}

\]
where

c

i
j

=
−

c

j
i

{c_{ij}} = - {c_{ji}}

and “

⋅
\cdot

” is the multiplication in

A
+

{A^ + }

. This applies in particular to simple nodal noncommutative Jordan algebras (of characteristic not 2). These results suggest a method of constructing noncommutative Jordan algebras of the given form. We have done this with the restriction that the

c

i
j

{c_{ij}}

lie in F1. The last result is that if A is a finitely generated simple noncommutative algebra of characteristic 0 of this form, then Der (A) is an infinite simple Lie algebra of a known type.
Local theory of complex functional differential equations
Authors: Oberg Robert J. 
Published: July 2010

We consider the equation

(
∗

)

f
′

(
z
)
=
F
(
z
,
f
(
z
)
,
f
(
g
(
z
)
)
)

( ^\ast )f’(z) = F(z,f(z),f(g(z)))

where

F
(
z
,
u
,
w
)

F(z,u,w)

and

g
(
z
)

g(z)

are given analytic functions and

f
(
z
)

f(z)

is an unknown function. The question of local existence of a solution of

(
∗

)

( ^\ast )

about a point

z
0

{z_0}

is natural only if

g
(

z
0

)
=

z
0

g({z_0}) = {z_0}

. We classify fixed points

z
0

{z_0}

of g as attractive if

|

g
′

(

z
0

)

|

>
1

| {g’({z_0})} | > 1

, indifferent if

|

g
′

(

z
0

)

|

=
1

| {g’({z_0})} | = 1

, and repulsive if

|

g
′

(

z
0

)

|

>
1

| {g’({z_0})} | > 1

. In the attractive case

(
∗

)

( ^\ast )

has a unique analytic solution satisfying an initial condition

f
(

z
0

)
=

w
0

f({z_0}) = {w_0}

. This solution depends continuously on

w
0

{w_0}

and on the functions F and g. For “most” indifferent fixed points the initial-value problem has a unique solution. Around a repulsive fixed point a solution in general does not exist, though in exceptional cases there may exist a singular solution which disappears if the equation is subjected to a suitable small perturbation.
Some new classes of kernels whose Fredholm determinants have order less than one
Authors: Swann Dale W. 
Published: July 2010

Let

K
(
s
,
t
)

K(s,t)

be a complex-valued

L
2

{L_2}

kernel on the square

a
≦
s
,
t
≦
b

a \leqq s,t \leqq b

and

{

λ
v

}

\{ {\lambda _v}\}

, perhaps empty, denote the set of finite characteristic values (f.c.v.) of

K
K

, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation

ϕ
v

(
s
)
=

λ
v

∫
a
b

K
(
s
,
t
)

ϕ
v

(
t
)
d
t

{\phi _v}(s) = {\lambda _v}\int _a^b {K(s,t){\phi _v}(t)dt}

, where the

ϕ
v

(
s
)

{\phi _v}(s)

are nontrivial

L
2

{L_2}

functions on

[
a
,
b
]

[a,b]

. Further let

k
1

=

∫
a
b

K
(
s
,
s
)

{k_1} = \int _a^b {K(s,s)}

be well defined so that the Fredholm determinant of

K
,
D
(
λ
)

K,D(\lambda )

, exists, and let

μ
\mu

be the order of this entire function. It is shown that (1) if

K
(
s
,
t
)

K(s,t)

is a function of bounded variation in the sense of Hardy-Krause, then

μ
≦
1

\mu \leqq 1

; (2) if in addition to the assumption (1),

K
(
s
,
t
)

K(s,t)

satisfies a uniform Lipschitz condition of order

α
>
0

\alpha > 0

with respect to either variable, then

μ
>
1

\mu > 1

and

k
1

=

Σ
v

1

/

λ
v

{k_1} = {\Sigma _v}1/{\lambda _v}

; (3) if

K
(
s
,
t
)

K(s,t)

is absolutely continuous as a function of two variables and

∂
2

K

/

∂
s
∂
t

{\partial ^2}K/\partial s\partial t

(which exists almost everywhere) belongs to class

L
p

{L_p}

for some

p
>
1

p > 1

, then

μ
>
1

\mu > 1

and

k
1

=

Σ
v

1

/

λ
v

{k_1} = {\Sigma _v}1/{\lambda _v}

. In (2) and (3), the condition

k
1

≠
0

{k_1} \ne 0

implies

K
(
s
,
t
)

K(s,t)

possesses at least one f.c.v.
Reflection principle for systems of first order elliptic equations with analytic coefficients
Authors: Yu Chung Ling 
Published: July 2010

Let T be a simply connected domain of the

z
=
x
+
i
y

z = x + iy

plane, whose boundary contains a portion

σ
\sigma

of the x-axis. Also let

A
(
z
,
ζ
)
,
B
(
z
,
ζ
)
,
F
(
z
,
ζ
)
,
α
(
z
)
,
β
(
z
)

A(z,\zeta ),B(z,\zeta ),F(z,\zeta ),\alpha (z),\beta (z)

and

ρ
(
z
)

\rho (z)

be holomorphic functions for

z
,
ζ
∈
T
∪
σ
∪

T
¯

z,\zeta \in T \cup \sigma \cup \bar T

, with

α
(
z
)
−
i
β
(
z
)
≠
0

\alpha (z) - i\beta (z) \ne 0

for

z
∈

T
¯

∪
σ
,
α
(
z
)
+
i
β
(
z
)
≠
0

z \in \bar T \cup \sigma ,\alpha (z) + i\beta (z) \ne 0

for

z
∈
T
∪
σ

z \in T \cup \sigma

. Furthermore, we assume that

α
(
x
)

\alpha (x)

and

β
(
x
)

\beta (x)

are real valued functions for

x
∈
σ

x \in \sigma

. Our reflection principle states that for any solution

w
=
u
+
i
v

w = u + iv

of an equation of the type

∂
w

/

∂

z
¯

=
A
(
z
,

z
¯

)
w
+
B
(
z
,

z
¯

)

w
¯

+
F
(
z
,

z
¯

)

\partial w/\partial \bar z = A(z,\bar z)w + B(z,\bar z)\bar w + F(z,\bar z)

in T under the boundary condition

α
(
x
)
u
+
β
(
x
)
v
=
ρ
(
x
)

\alpha (x)u + \beta (x)v = \rho (x)

on

σ
,
w

\sigma ,w

can be continued analytically across the x-axis, onto the entire mirror image

T
¯

\bar T

.
Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators
Authors: Farias Antonio O. 
Published: July 2010

A Titus transformation

T
=
⟨
α
,
v
⟩

T = \langle \alpha ,v\rangle

is a linear operator on the vector space of

C
∞

{C^\infty }

mappings from the circle into the plane given by

(
T
f
)
(
t
)
=
(
⟨
α
,
v
⟩
f
)
(
t
)
=
f
(
t
)
+
α
(
t
)
det
[
v
,

f
′

(
t
)
]
v

(Tf)(t) = (\langle \alpha ,v\rangle f)(t) = f(t) + \alpha (t)\det [v,f’(t)]v

, where

α
\alpha

is a nonnegative,

C
∞

{C^\infty }

function on the circle

S
1

{S^1}

. Let

τ
\tau

denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of

τ
\tau

of a degenerate curve,

α
0

v
0

{\alpha _0}{v_0}

, where

α
0

{\alpha _0}

is a

C
∞

{C^\infty }

function on

S
1

{S^1}

and

v
0

{v_0}

is fixed in the plane

R
2

{R^2}

. A

C
∞

{C^\infty }

mapping

f
:

S
1

→

R
2

f:{S^1} \to {R^2}

is called properly extendable if there is a

C
∞

{C^\infty }

mapping

F
:

D
−

→

R
2

F:{D^ - } \to {R^2}

, D the open unit disk and

D
−

{D^ - }

its closure, such that

J
F

≧
0

{J_F} \geqq 0

on

D
,

J
F

>
0

D,{J_F} > 0

near the boundary

S
1

{S^1}

of

D
−

{D^ - }

and

F

|

s
1

=
f

F{|_{{s^1}}} = f

. A

C
∞

{C^\infty }

mapping

f
:

S
1

→

R
2

f:{S^1} \to {R^2}

is called normal if it is an immersion with no triple points and all its double points are transversal. The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping. An application is made to a class of integral operators of the convolution type,

y
(
t
)
=
−

∫
0

2
π

k
(
s
)
x
(
t
−
s
)
d
s

y(t) = - \smallint _0^{2\pi } {k(s)x(t - s)ds}

. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert’s transform of potential theory,

y
(
t
)
=

∫
0

2
π

cot
⁡
(
s

/

2
)
x
(
t
−
s
)
d
s

y(t) = \smallint _0^{2\pi } {\cot (s/2)x(t - s)ds}

, which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.
Local behaviour of solutions of stochastic integral equations
Authors: Anderson William J. 
Published: July 2010

Let X denote the solution process of the stochastic equation

d
X
(
t
)
=
a
(
X
(
t
)
)
d
t
+
σ
(
X
(
t
)
)
d
W
(
t
)

dX(t) = a(X(t))dt + \sigma (X(t))dW(t)

. In this paper, conditions on

a
(
⋅
)

a( \cdot )

and

σ
(
⋅
)

\sigma ( \cdot )

are given under which the sample paths of X are differentiate at

t
=
0

t = 0

with probability one. Variations of these results are obtained leading to a new uniqueness criterion for solutions of stochastic equations. If

σ
(
⋅
)

\sigma ( \cdot )

is Hölder continuous with exponent greater than

1
2

\tfrac {1}{2}

and

a
(
⋅
)

a( \cdot )

satisfies a Lipschitz condition, it is shown that in the one-dimensional case the above equation has only one continuous solution.
The variation of singular cycles in an algebraic family of morphisms
Authors: Roberts Joel 
Published: July 2010

(1) Let

g
:

V
r

→

W
m

(
m
≧
r
)

g:{V^r} \to {W^m}(m \geqq r)

be a morphism of nonsingular varieties over an algebraically closed field. Under certain conditions, one can define a cycle

S
i

{S_i}

on

V
V

with

Supp
⁡
(

S
i

)
=
{
x

|

dim

k
(
x
)

(

Ω

X

/

Y

1

)
(
x
)
≧
i
}

\operatorname {Supp} ({S_i}) = \{ x|{\dim _{k(x)}}(\Omega _{X/Y}^1)(x) \geqq i\}

. The multiplicity of a component of

S
i

{S_i}

can be computed directly from local equations for

g
g

. If

V
r

⊂

P
n

{V^r} \subset {P^n}

, and if

g
:
V
→

P
m

g:V \to {P^m}

is induced by projection from a suitable linear subspace of

P
n

{P^n}

, then

S
1

{S_1}

is

c

m
−
r
+
1

(
N
⊗

O

(
−
1
)
)

{c_{m - r + 1}}(N \otimes \mathcal {O}( - 1))

, up to rational equivalence, where

N
N

is the normal bundle of

V
V

in

P
n

{P^n}

. (2) Let

f
:
X
→
S

f:X \to S

be a smooth projective morphism of noetherian schemes, where

S
S

is connected, and the fibres of

f
f

are absolutely irreducible

r
r

-dimensional varieties. For a geometric point

η
:
Spec
⁡
(
k
)
→
S

\eta :\operatorname {Spec} (k) \to S

, and a locally free sheaf

E
E

on

X
X

, let

X
η

{X_\eta }

be the corresponding geometric fibre, and

E
η

{E_\eta }

the sheaf induced on

X
η

{X_\eta }

. If

E
1

,
…
,

E
m

{E_1}, \ldots ,{E_m}

are locally free sheaves on

X
X

, and if

i
1

+
⋯
+

i
m

=
r

{i_1} + \cdots + {i_m} = r

, then the degree of the zero-cycle

c

i
1

(

E

1
η

)
⋯

c

i
m

(

E

m
η

)

{c_{{i_1}}}({E_{1\eta }}) \cdots {c_{{i_m}}}({E_{m\eta }})

is independent of the choice of

η
\eta

. (3) The results of (1) and (2) are used to study the behavior under specialization of a closed subvariety

V
′

⊂

P

2
r
−
1

V’ \subset {P^{2r - 1}}

which is the image under generic projection of a nonsingular

V
r

⊂

P
n

{V^r} \subset {P^n}

.
On trigonometric series associated with separable, translation invariant subspaces of 𝐿^{∞}(𝐺)
Authors: Blei Ron C. 
Published: July 2010

G
G

denotes a compact abelian group, and

Γ
\Gamma

denotes its dual. Our main result is that every non-Sidon set

E
⊂
Γ

E \subset \Gamma

contains a non-Sidon set

F
F

such that

L
F
∞

(
G
)
=

⊕
l

1

i
=
1

∞

C

F
i

(
G
)

L_F^\infty (G) = { \oplus _l}1_{i = 1}^\infty {C_{{F_i}}}(G)

, where the

F
i

{F_i}

’s are finite, mutually disjoint, and

∪

i
=
1

∞

F
i

=
F

\cup _{i = 1}^\infty {F_i} = F

.
Simple groups of order 2^{𝑎}3^{𝑏}5^{𝑐}7^{𝑑}𝑝
Authors: Alex Leo J. 
Published: July 2010

Let

PSL

(
n
,
q
)

{\operatorname {PSL}}(n,q)

denote the projective special linear group of degree

n
n

over

GF

(
q
)

{\text {GF}}(q)

, the field with

q
q

elements. The following theorem is proved. Theorem. Let

G
G

be a simple group of order

2
a

3
b

5
c

7
d

p
,
a
>
0
,
p

{2^a}{3^b}{5^c}{7^d}p,a > 0,p

an odd prime. If the index of a Sylow

p
p

-subgroup of

G
G

in its normalizer is two, then

G
G

is isomorphic to one of the groups,

PSL

(
2
,
5
)
,

PSL

(
2
,
7
)
,

PSL

(
2
,
9
)
,

PSL

(
(
2
,
8
)
,

PSL

(
2
,
16
)
,

PSL

(
2
,
25
)
,

PSL

(
2
,
27
)
,

PSL

(
2
,
81
)

{\operatorname {PSL}}(2,5),{\operatorname {PSL}}(2,7),{\operatorname {PSL}}(2,9),{\operatorname {PSL}}((2,8),{\operatorname {PSL}}(2,16),{\operatorname {PSL}}(2,25),{\operatorname {PSL}}(2,27),{\operatorname {PSL}}(2,81)

, and

PSL

(
3
,
4
)

{\operatorname {PSL}}(3,4)

.
Interpolation between consecutive conjugate points of an 𝑛th order linear differential equation
Authors: Gustafson G. B. 
Published: July 2010

The interpolation problem

x

(
n
)

+

P

n
−
1

x

(
n
−
1
)

+
⋯
+

P
0

x
=
0

{x^{(n)}} + {P_{n - 1}}{x^{(n - 1)}} + \cdots + {P_0}x = 0

,

x

(
i
)

(

t
j

)
=
0
,
i
=
0
,
⋯
,

k
j

−
1
,
j
=
0
,
⋯
,
m

{x^{(i)}}({t_j}) = 0,i = 0, \cdots ,{k_j} - 1,j = 0, \cdots ,m

, is studied on the conjugate interval

[
a
,

η
1

(
a
)
]

[a,{\eta _1}(a)]

. The main result is that there exists an essentially unique nontrivial solution of the problem almost everywhere, provided

k
1

+
⋯
+

k
m

≥
n

{k_1} + \cdots + {k_m} \geq n

, and cer tain other inequalities are satisfied, with

a
=

t
0

>

t
1

>
⋯
>

t
m

=

η
1

(
a
)

a = {t_0} > {t_1} > \cdots > {t_m} = {\eta _1}(a)

. In particular, this paper corrects the results of Azbelev and Caljuk (Mat. Sb. 51 (93) (1960), 475-486; English transl., Amer. Math. Soc. Transl. (2) 42 (1964), 233-245) on third order equations, and shows that their results are correct almost everywhere.
Representations and classifications of stochastic processes
Authors: Johnson Dudley Paul 
Published: July 2010

We show that to every stochastic process X one can associate a unique collection

(
Φ
,

Φ
+

,
T
(
t
)
,
E
(
U
)
,

p
∗

)

(\Phi ,{\Phi _ + },T(t),E(U),{p^\ast })

consisting of a linear space

Φ
\Phi

, on which is defined a linear functional

p
∗

{p^ \ast }

, together with a convex subset

Φ
+

{\Phi _ + }

which is invariant under the semigroup of operators

T
(
t
)

T(t)

and the resolution of the identity

E
(
U
)

E(U)

. The joint distributions of X, there being one version for each

ϕ
∈

Φ
+

\phi \in {\Phi _ + }

, are then given by
\[

P
ϕ

(
X
(

t
1

)
∈

U
1

,
⋯
,
X
(

t
1

+
⋯
+

t
n

)
∈

U
n

)
=

p
∗

E
(

U
n

)
T
(

t
n

)
⋯
E
(

U
1

)
T
(

t
1

)
ϕ
.

{P_\phi }(X({t_1}) \in {U_1}, \cdots ,X({t_1} + \cdots + {t_n}) \in {U_n}) = {p^ \ast }E({U_n})T({t_n}) \cdots E({U_1})T({t_1})\phi .

\]
To each

ϕ
\phi

contained in the extreme points

Φ

+
+

{\Phi _{ + + }}

of

Φ
+

{\Phi _ + }

and each time t we find a probability measure

P
t
∗

(
ϕ
,
⋅
)

P_t^ \ast (\phi , \cdot )

on

Φ

+
+

{\Phi _{ + + }}

such that

T
(
t
)
ϕ
=

∫

Φ

+
+

ψ

P
t
∗

(
ϕ
,
d
ψ
)

T(t)\phi = {\smallint _{{\Phi _{ + + }}}}\psi P_t^ \ast (\phi ,d\psi )

.

P
t
∗

P_t^ \ast

is the transition probability function of a temporally homogeneous Markov process

X
∗

{X^ \ast }

on

Φ

+
+

{\Phi _{ + + }}

for which there exists a function f such that

X
=
f
(

X
∗

)

X = f({X^ \ast })

. We show that in a certain sense

X
∗

{X^ \ast }

is the smallest of all Markov processes Y for which there exists a function g with

X
=
g
(
Y
)

X = g(Y)

. We then apply these results to a class of stochastic process in which future and past are independent given the present and the conditional distribution, on the past, of a collection of random variables in the future.
On Fourier transforms
Authors: Nasim C. 
Published: July 2010

If

f
(
x
)

f(x)

and

g
(
x
)

g(x)

satisfy the equations
\[

g
(
x
)
=

d

d
x

∫
0
∞

1
t

f
(
t
)

k
1

(
x
t
)
d
t
,

f
(
x
)
=

d

d
x

∫
0
∞

1
t

g
(
t
)

k
1

(
x
t
)
d
t
,

g(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}f(t){k_1}(xt)dt,\quad f(x) = \frac {d}{{dx}}\int _0^\infty \frac {1}{t}g(t){k_1}(xt)dt,

\]
then we call f and g a pair of

k
1

{k_1}

-transforms, where
\[

k
1

=

1

2
π
i

∫

1

/

2
−
i
∞

1

/

2
+
i
∞

K
(
s
)

1
−
s

x

1
−
s

d
s
.

k_1 = \frac {1}{2\pi i} \int _{1/2 - i\infty }^{1/2 + i\infty } \frac {K(s)}{1 - s} x^{1-s}\,ds.

\]
In this paper alternative sets of conditions are established for f and g to be

k
1

{k_1}

-transform provided

K
(
s
)

K(s)

is decomposable in a special way. These conditions involve simpler functions, which replace the kernel

k
1

(
x
)

{k_1}(x)

. Results are proved for the function spaces

L
2

{L^2}

. The necessary and sufficient conditions are established for the two functions to be self-reciprocal. Conditions are given for generating pairs of transforms for a given kernel. Two examples are given at the end to illustrate the methods and the advantage of the results.
Fixed point iterations using infinite matrices
Authors: Rhoades B. E. 
Published: July 2010

Let E be a closed, bounded, convex subset of a Banach space

X
,
f
:
E
→
E

X,f:E \to E

. Consider the iteration scheme defined by

x
¯

0

=

x
0

∈
E
,

x
¯

n
+
1

=
f
(

x
n

)
,

x
n

=

Σ

k
=
0

n

a

n
k

x
¯

k

,

n
≥
1

{\bar x_0} = {x_0} \in E,{\bar x_{n + 1}} = f({x_n}),{x_n} = \Sigma _{k = 0}^n{a_{nk}}{\bar x_k},\;n \geq 1

, where A is a regular weighted mean matrix. For particular spaces X and functions f we show that this iterative scheme converges to a fixed point of f.
The inertial aspects of Stein’s condition 𝐻-𝐶*𝐻𝐶≫𝑂
Authors: Cain Bryan E. 
Published: July 2010

To each bounded operator C on the complex Hilbert space

H

\mathcal {H}

we associate the vector space

K

C

{\mathcal {K}_C}

consisting of those

x
∈

H

x \in \mathcal {H}

for which

C
n

x
→
0

{C^n}x \to 0

as

n
→
∞

n \to \infty

. We let

α
(
C
)

\alpha (C)

denote the dimension of the closure of

K

C

{\mathcal {K}_C}

and we set

β
(
C
)
=
dim
⁡
(

K

C
⊥

)

\beta (C) = \dim (\mathcal {K}_C^ \bot )

. Our main theorem states that if H is Hermitian and if

H
−

C
∗

H
C

H - {C^ \ast }HC

is positive and invertible then

α
(
C
)
≤
π
(
H
)
,
β
(
C
)
=
ν
(
H
)

\alpha (C) \leq \pi (H),\beta (C) = \nu (H)

, and

β
(
C
)
≥
δ
(
H
)

\beta (C) \geq \delta (H)

where

(
π
(
H
)
,
ν
(
H
)
,
δ
(
H
)
)

(\pi (H),\nu (H),\delta (H))

is the inertia of H. (That is,

π
(
H
)
=
dim

(

Range

E
[
(
0
,
∞
)
]
)

\pi (H) = \dim \;({\text {Range}}\;E[(0,\infty )])

) where E is the spectral measure of H;

ν
(
H
)
=
π
(
−
H
)

\nu (H) = \pi ( - H)

; and

δ
(
H
)
=
dim
⁡
(

Ker

H
)

\delta (H) = \dim ({\operatorname {Ker}}\;H)

.) We also show (l) that in general no stronger conclusion is possible, (2) that, unlike previous inertia theorems, our theorem allows 1 to lie in

σ
(
C
)

\sigma (C)

, the spectrum of C, and (3) that the main inertial results associated with the hypothesis that

Re
⁡
(
H
A
)

\operatorname {Re} (HA)

is positive and invertible can be derived from our theorem. Our theorems (1) characterize C in the extreme cases that either

π
(
H
)
=
0

\pi (H) = 0

or

ν
(
H
)
=
0

\nu (H) = 0

, and (2) prove that

α
(
C
)
=
π
(
H
)
,
β
(
C
)
=
ν
(
H
)
,
δ
(
H
)
=
0

\alpha (C) = \pi (H),\beta (C) = \nu (H),\delta (H) = 0

if either

1
∉
σ
(
C
)

1 \notin \sigma (C)

or

β
(
C
)
>
∞

\beta (C) > \infty

.
Extremal problems of distance geometry related to energy integrals
Authors: Alexander Ralph , Stolarsky Kenneth B. 
Published: July 2010

Let K be a compact set,

M

\mathcal {M}

a prescribed family of (possibly signed) Borel measures of total mass one supported by K, and f a continuous real-valued function on

K
×
K

K \times K

. We study the problem of determining for which

μ
∈

M

\mu \in \mathcal {M}

(if any) the energy integral

I
(
K
,
μ
)
=

∫
K

∫
K

f
(
x
,
y
)
d
μ
(
x
)
d
μ
(
y
)

I(K,\mu ) = \smallint _K {\smallint _K {f(x,y)d\mu (x)d\mu (y)} }

is maximal, and what this maximum is. The more symmetry K has, the more we can say; our results are best when K is a sphere. In particular, when

M

\mathcal {M}

is atomic we obtain good upper bounds for the sums of powers of all

(
2
n

)

(_2^n)

distances determined by n points on the surface of a sphere. We make use of results from Schoenberg’s theory of metric embedding, and of techniques devised by Pólya and Szegö for the calculation of transfinite diameters.
The homotopy type of the space of diffeomorphisms. I
Authors: Burghelea Dan , Lashof Richard 
Published: July 2010

A new proof is given of the unpublished results of Morlet on the relation between the homeomorphism group and the diffeomorphism group of a smooth manifold. In particular, the result

Diff

(

D
n

,
∂
)
≃

Ω

n
+
1

(

Top
n

/

O
n

)

{\operatorname {Diff}}({D^n},\partial ) \simeq {\Omega ^{n + 1}}({\text {Top}_n}/{O_n})

is obtained. The main technique is fibrewise smoothing.
Interpolation in a classical Hilbert space of entire functions
Authors: Young Robert M. 
Published: July 2010

Let H denote the Paley-Wiener space of entire functions of exponential type

π
\pi

which belong to

L
2

(
−
∞
,
∞
)

{L^2}( - \infty ,\infty )

on the real axis. A sequence

{

λ
n

}

\{ {\lambda _n}\}

of distinct complex numbers will be called an interpolating sequence for H if

T
H
⊃

l
2

TH \supset {l^2}

, where T is the mapping defined by

T
f
=
{
f
(

λ
n

)
}

Tf = \{ f({\lambda _n})\}

. If in addition

{

λ
n

}

\{ {\lambda _n}\}

is a set of uniqueness for H, then

{

λ
n

}

\{ {\lambda _n}\}

is called a complete interpolating sequence. The following results are established. If

Re
⁡
(

λ

n
+
1

)
−
Re
⁡
(

λ
n

)
≥
γ
>
1

\operatorname {Re} ({\lambda _{n + 1}}) - \operatorname {Re} ({\lambda _n}) \geq \gamma > 1

and if the imaginary part of

λ
n

{\lambda _n}

is sufficiently small, then

{

λ
n

}

\{ {\lambda _n}\}

is an interpolating sequence. If

|

Re
⁡
(

λ
n

)
−
n

|

≤
L
≤
(
log
⁡
2
)

/

π

(
−
∞
>
n
>
∞
)

|\operatorname {Re} ({\lambda _n}) - n| \leq L \leq (\log 2)/\pi \;( - \infty > n > \infty )

and if the imaginary part of

λ
n

{\lambda _n}

is uniformly bounded, then

{

λ
n

}

\{ {\lambda _n}\}

is a complete interpolating sequence and

{

e

i

λ
n

t

}

\{ {e^{i{\lambda _n}t}}\}

is a basis for

L
2

(
−
π
,
π
)

{L^2}( - \pi ,\pi )

. These results are used to investigate interpolating sequences in several related spaces of entire functions of exponential type.
Density of parts of algebras on the plane
Authors: O’Farrell Anthony G. 
Published: July 2010

We study the Gleason parts of a uniform algebra A on a compact subset of the plane, where it is assumed that for each point

x
∈

C

x \in {\text {C}}

the functions in A which are analytic in a neighborhood of x are uniformly dense in A. We prove that a part neighborhood N of a nonpeak point x for A satisfies a density condition of Wiener type at

x
:

Σ

n
=
1

+
∞

2
n

C
(

A
n

(
x
)
∖
N
)
>
+
∞

x:\Sigma _{n = 1}^{ + \infty }{2^n}C({A_n}(x)\backslash N) > + \infty

, and if A admits a pth order bounded point derivation at x, then N satisfies a stronger density condition:

Σ

n
=
1

+
∞

2

(
p
+
1
)
n

C
(

A
n

(
x
)
∖
N
)
>
+
∞

\Sigma _{n = 1}^{ + \infty }{2^{(p + 1)n}}C({A_n}(x)\backslash N) > + \infty

. Here C is Newtonian capacity and

A
n

(
x
)

{A_n}(x)

is

{
z
∈

C

:

2

−
n
−
1

≤

|

z
−
x

|

≤

2

−
n

}

\{ z \in {\text {C}}:{2^{ - n - 1}} \leq |z - x| \leq {2^{ - n}}\}

. These results strengthen and extend Browder’s metric density theorem. The relation with potential theory is examined, and analogous results for the algebra

H
∞

(
U
)

{H^\infty }(U)

are obtained as corollaries.