bluuebunny/crossref_metadata_2025_split huggingface dataset:
Weakly starlike meromorphic univalent functions
Authors: Libera Richard J. , Livingston Albert E. 
Published: July 2010

A weakly starlike meromorphic univalent function is one of the form

f
(
z
)
=
−
ρ
z
g
(
z
)

[
(
z
−
ρ
)
(
1
−
ρ
z
)

]

−
1

f(z) = - \rho zg(z){[(z - \rho )(1 - \rho z)]^{ - 1}}

for

0
>
ρ
>
1

0 > \rho > 1

and

g
(
z
)

g(z)

a meromorphic starlike function. The behavior of coefficients and growth of this class of functions and of a subset are studied.
Results on sums of continued fractions
Authors: Hlavka James L. 
Published: July 2010

Let

F
(
m
)

F(m)

be the (Cantor) set of infinite continued fractions with partial quotients no greater than m and let

F
(
m
)
+
F
(
n
)
=
{
α
+
β
:
α
∈
F
(
m
)
,
β
∈
F
(
n
)
}

F(m) + F(n) = \{ \alpha + \beta :\alpha \in F(m),\beta \in F(n)\}

. We show that

F
(
3
)
+
F
(
4
)

F(3) + F(4)

is an interval of length 1.14 ... so every real number is the sum of an integer, an element of

F
(
3
)

F(3)

and an element of

F
(
4
)

F(4)

. Similar results are given for

F
(
2
)
+
F
(
7
)
,
F
(
2
)
+
F
(
2
)
+
F
(
4
)
,
F
(
2
)
+
F
(
3
)
+
F
(
3
)

F(2) + F(7),F(2) + F(2) + F(4),F(2) + F(3) + F(3)

and

F
(
2
)
+
F
(
2
)
+
F
(
2
)
+
F
(
2
)

F(2) + F(2) + F(2) + F(2)

. The techniques used are applicable to any Cantor sets in R for which certain parameters can be evaluated.
Gelfond’s method for algebraic independence
Authors: Brownawell W. Dale 
Published: July 2010

This paper extends Gelfond’s method for algebraic independence to fields

K
K

with transcendence type

⩽
τ

\leqslant \tau

. The main results show that the elements of a transcendence basis for

K
K

and at least two more numbers from a prescribed set are algebraically independent over

Q
Q

. The theorems have a common hypothesis:

{

α
1

,
…
,

α
M

}
,
{

β
1

,
…
,

β
N

}

\{ {\alpha _1}, \ldots ,{\alpha _M}\} ,\{ {\beta _1}, \ldots ,{\beta _N}\}

are sets of complex numbers, each of which is

Q
Q

-linearly independent. THEOREM A. If

(
2
τ
−
1
)
>
M
N

(2\tau - 1) > MN

, then at least two of the numbers

α
i

,

β
j

,
exp
⁡
(

α
i

β
j

)
,
1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

{\alpha _i},{\beta _j},\exp ({\alpha _i}{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

. THEOREM B. If

2
τ
(
M
+
N
)
⩽
M
N
+
M

2\tau (M + N) \leqslant MN + M

, then at least two of the numbers

α
i

,
exp
⁡
(

α
i

,

β
j

)
,
1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

{\alpha _i},\exp ({\alpha _i},{\beta _j}),1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

. THEOREM C. If

2
τ
(
M
+
N
)
⩽
M
N

2\tau (M + N) \leqslant MN

, then at least two of the numbers

1
⩽
i
⩽
M
,
1
⩽
j
⩽
N

1 \leqslant i \leqslant M,1 \leqslant j \leqslant N

, are algebraically dependent over

K
K

.
The predual theorem to the Jacobson-Bourbaki theorem
Authors: Sweedler Moss 
Published: July 2010

Suppose

R

→

φ

S

R\xrightarrow {\varphi }S

is a map of rings. S need not be an R algebra since R may not be commutative. Even if R is commutative it may not have central image in S. Nevertheless the ring structure on S can be expressed in terms of two maps
\[

S

⊗
R

S

→

(

s
1

⊗

s
2

→

s
1

s
2

)

S
,

R

→

φ

S
,

S{ \otimes _R}S\xrightarrow {{({s_1} \otimes {s_2} \to {s_1}{s_2})}}S,\quad R\xrightarrow {\varphi }S,

\]
which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an R-coring. Suppose R is an overing of B. Let

C
B

=
R

⊗
B

R

{C_B} = R{ \otimes _B}R

. There are maps
\[

C
B

=
R

⊗
B

R

→

(

r
1

⊗

r
2

→

r
1

⊗
1
⊗

r
2

)

R

⊗
B

R

⊗
B

R
=
(

C
B

)

⊗
R

(

C
B

)
,

C
B

=
R

⊗
B

R

→

(

r
1

⊗

r
2

→

r
1

r
2

)

R
.

\begin {array}{*{20}{c}} {{C_B} = R{ \otimes _B}R\xrightarrow {{({r_1} \otimes {r_2} \to {r_1} \otimes 1 \otimes {r_2})}}R{ \otimes _B}R{ \otimes _B}R = ({C_B}){ \otimes _R}({C_B}),} \hfill \\ {{C_B} = R{ \otimes _B}R\xrightarrow {{({r_1} \otimes {r_2} \to {r_1}{r_2})}}R.} \hfill \\ \end {array}

\]
These maps give

C
B

{C_B}

an R-coring structure. The dual

∗

C
B

^\ast {C_B}

is naturally isomorphic to the ring

End

B
−

R

{\text {End}_{{B^ - }}}R

of B-linear endomorphisms of R considered as a left B-module. In case B happens to be the subring of R generated by 1, we write

C

Z

{C_{\text {Z}}}

. Then

∗

C

Z

^\ast {C_{\text {Z}}}

is

End

Z

R

{\text {End}_{\text {Z}}}R

, the endomorphism ring of R considered as an additive group. This gives a clue how certain R-corings correspond to subrings of R and subrings of

End

Z

R

{\text {End}_{\text {Z}}}R

, both major ingredients of the Jacobson-Bourbaki theorem.

1
⊗
1

1 \otimes 1

is a “grouplike” element in the R-coring

C

Z

{C_{\text {Z}}}

(and should be thought of as a generic automorphism of R). Suppose R is a division ring and B a subring which is a division ring. The natural map

C

Z

→

C
B

{C_{\text {Z}}} \to {C_B}

is a surjective coring map. Conversely if

C

Z

→

π

D

{C_{\text {Z}}}\xrightarrow {\pi }D

is a (surjective) coring map then

π
(
1
⊗
1
)

\pi (1 \otimes 1)

is a grouplike in D and

{
r
∈
R

|

r
π
(
1
⊗
1
)
=
π
(
1
⊗
1
)
r
}

\{ r \in R|r\pi (1 \otimes 1) = \pi (1 \otimes 1)r\}

is a subring of R which is a division ring. This gives a bijective correspondence between the quotient corings of

C

⊗

Z

C

C{ \otimes _{\text {Z}}}C

and the subrings of R which are division rings. We show how the Jacobson-Bourbaki correspondence is dual to the above correspondence.
Convolution, differential equations, and entire functions of exponential type
Authors: Mugler Dale H. 
Published: July 2010

The Whittaker-Shannon interpolation formula, or “cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients

Σ

n
=
0

∞

a
n

f

(
n
)

(
z
)
=
∫
f
(
z
−
t
)
d
μ
(
t
)

\Sigma _{n = 0}^\infty {a_n}{f^{(n)}}(z) = \smallint f(z - t)d\mu (t)

where the integral is taken over the whole real line with respect to the measure

μ
\mu

. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity

f
(
z
)
=
f
(
z
+
τ
)

f(z) = f(z + \tau )

. Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.
Evolution system approximations of solutions to closed linear operator equations
Authors: Purdom Seaton D. 
Published: July 2010

With S a linearly ordered set with the least upper bound property, with g a nonincreasing real-valued function on S, and with A a densely defined dissipative linear operator, an evolution system M is developed to solve the modified Stieljes integral equation

M
(
s
,
t
)
x
=
x
+
A
(
(
L
)

∫
S
t

d
g
M
(
⋅
,
t
)
x
)

M(s,t)x = x + A((L)\smallint _S^t dgM( \cdot ,t)x)

. An affine version of this equation is also considered. Under the hypothesis that the evolution system associated with the linear equation is strongly (resp. weakly) asymptotically convergent, an evolution system is used to approximate strongly (resp. weakly) solutions to the closed operator equation

A
y
=
−
z

Ay = - z

.
Involutions on homotopy spheres and their gluing diffeomorphisms
Authors: Liang Chao Chu 
Published: July 2010

Let

h
S
(

P

2
n
+
1

)

hS({P^{2n + 1}})

denote the set of equivalence classes of smooth fixed-point free involutions on

(
2
n
+
1
)

(2n + 1)

-dimensional homotopy spheres. Browder and Livesay defined an invariant

σ
(

Σ

2
n
+
1

,
T
)

\sigma ({\Sigma ^{2n + 1}},T)

for each

(

Σ

2
n
+
1

,
T
)
∈
h
S
(

P

2
n
+
1

)

({\Sigma ^{2n + 1}},T) \in hS({P^{2n + 1}})

, where

σ
∈
Z

\sigma \in Z

if n is odd,

σ
∈

Z
2

\sigma \in {Z_2}

if n is even. They showed that for

n
⩾
3
,
σ
(

Σ

2
n
+
1

,
T
)
=
0

n \geqslant 3,\sigma ({\Sigma ^{2n + 1}},T) = 0

if and only if

(

Σ

2
n
+
1

,
T
)

({\Sigma ^{2n + 1}},T)

admits a codim 1 invariant sphere. For any

(

Σ

2
n
+
1

,
T
)

({\Sigma ^{2n + 1}},T)

, there exists an A-equivariant diffeomorphism f of

S
n

×

S
n

{S^n} \times {S^n}

such that

(

Σ

2
n
+
1

,
T
)
=
(

S
n

×

D

n
+
1

,
A
)

∪
f

(

D

n
+
1

×

S
n

,
A
)

({\Sigma ^{2n + 1}},T) = ({S^n} \times {D^{n + 1}},A){ \cup _f}({D^{n + 1}} \times {S^n},A)

, where A denotes the antipodal map. Let

β
(
f
)
=
σ
(

Σ

2
n
+
1

,
T
)

\beta (f) = \sigma ({\Sigma ^{2n + 1}},T)

. In the case n is odd, we can show that the Browder-Livesay invariant is additive:

β
(
f
g
)
=
β
(
f
)
+
β
(
g
)

\beta (fg) = \beta (f) + \beta (g)

. But if n is even, then there exists f and g such that

β
(
g
f
)
=
β
(
g
)
+
β
(
f
)
≠
β
(
f
g
)

\beta (gf) = \beta (g) + \beta (f) \ne \beta (fg)

. Let

D
0

(

S
n

×

S
n

,
A
)

{D_0}({S^n} \times {S^n},A)

be the group of concordance classes of A-equivariant diffeomorphisms which are homotopic to the identity map of

S
n

×

S
n

{S^n} \times {S^n}

. We can prove that “For

n
≡
0
,
1
,
2
mod
4
,
h
S
(

P

2
n
+
1

)

n \equiv 0,1,2 \bmod 4, hS({P^{2n + 1}})

is in 1-1 correspondence with a subgroup of

D
0

(

S
n

×

S
n

,
A
)

{D_0}({S^n} \times {S^n},A)

. As an application of these theorems, we demonstrated that “Let

Σ
0

8
k
+
3

\Sigma _0^{8k + 3}

denote the generator of

b

P

8
k
+
4

b{P_{8k + 4}}

. Then the number of

(

Σ
0

8
k
+
3

,
T
)

(\Sigma _0^{8k + 3},T)

’s with

σ
(

Σ
0

8
k
+
3

,
T
)
=
0

\sigma (\Sigma _0^{8k + 3},T) = 0

is either 0 or equal to the number of

(

S

8
k
+
3

,
T
)

({S^{8k + 3}},T)

’s with

σ
(

S

8
k
+
3

,
T
)
=
0

\sigma ({S^{8k + 3}},T) = 0

, where

S

8
k
+
3

{S^{8k + 3}}

denotes the standard sphere".
Moduli of continuity for exponential Lipschitz classes
Authors: De Land Paul 
Published: July 2010

Let

Ψ
\Psi

be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to

Ψ
\Psi

be given as
\[

Q
Ψ

(
δ
,
f
)
=
inf

{

λ
:

1
δ

∬

|

x
−
y

|

⩽
δ

Ψ

(

|

f
(
x
)
−
f
(
y
)

|

λ

)

d
x

d
y

⩽
Ψ
(
1
)

}

.

{Q_\Psi }(\delta ,f) = \inf \left \{ {\lambda :\frac {1}{\delta }\iint \limits _{|x - y| \leqslant \delta } {\Psi \left ( {\frac {{|f(x) - f(y)|}}{\lambda }} \right )}\;dx\;dy\; \leqslant \Psi (1)} \right \}.

\]
It is shown that

∫
0
1

Q
Ψ

(
δ
,
f
)

d

(
−

Ψ

−
1

(
c

/

δ
)
)
>
∞

\smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) > \infty

guarantees the essential continuity of f, and, in fact, a uniform Lipschitz estimate is given. In the case that

Ψ
(
u
)
=
exp

u
2

\Psi (u) = \exp \;{u^2}

the above condition reduces to
\[

∫
0
1

Q

exp

u
2

(
δ
,
f
)

d
δ

δ

log
⁡
(
c

/

δ
)

>
∞
.

\int _0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac {{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; > \infty .}

\]
This exponential square condition is satisfied almost surely by the random Fourier series

f
t

(
x
)
=

Σ

n
=
1

∞

a
n

R
n

(
t
)

e

i
n
x

{f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}

, where

{

R
n

}

\{ {R_n}\}

is the Rademacher system, as long as
\[

∫
0
1

a
n
2

sin

2

(
n
δ

/

2
)

d
δ

δ

log
⁡
(
1

/

δ
)

>
∞
.

\int _0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac {{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; > \infty .}

\]
Hence, the random essential continuity of

f
t

(
x
)

{f_t}(x)

is guaranteed by each of the above conditions.
Higher algebraic 𝐾-theories
Authors: Anderson D. , Karoubi M. , Wagoner J. 
Published: July 2010

A homotopy fibration is established relating the Volodin or BN-pair definition of algebraic K-theory to the theory defined by Quillen. In [2] we outlined the construction of natural homomorphisms
\[

K
∗
Q

→

K
∗

B
N

→

K
∗
V

→

K
∗

K
V

K_ \ast ^Q \to K_ \ast ^{BN} \to K_ \ast ^V \to K_ \ast ^{KV}

\]
between higher algebraic K-theories

K
∗
Q

K_ \ast ^Q

of [10] and [11],

K
∗

B
N

K_ \ast ^{BN}

of [17],

K
∗
V

K_ \ast ^V

of [16], and

K
∗

K
V

K_ \ast ^{KV}

of [7] and [8]. This was one of the steps in proving the various definitions of higher K-theory are equivalent. It turns out they all agree-including the theory

K
∗
S

K_ \ast ^S

of [14], [5], and [8]-provided one restricts to the category of regular rings when using

K
∗

K
V

K_\ast ^{KV}

. See [1], [2], [5], [8] and [18]. The purpose of this paper is to prove the following theorem, announced in [2], which yields the construction of

K
∗
Q

→

K
∗

B
N

K_ \ast ^Q \to K_ \ast ^{BN}

. Theorem. For any associative ring with identity A
\[

G

L

B
N

(
A
)
→
B

{

U
F

}
+

→
B
G
L

(
A

)
+

G{L^{BN}}(A) \to B{\{ {U_F}\} ^ + } \to BGL{(A)^ + }

\]
is a homotopy fibration. For the reader’s convenience and because the presentation of the BN-pair K-theory

K
∗

B
N

K_ \ast ^{BN}

used here is slightly different from that of [17], we shall briefly recall the definition of

G

L

B
N

G{L^{BN}}

and

B
{

U
F

}

B\{ {U_F}\}

in the first section.
Diffeomorphisms almost regularly homotopic to the identity
Authors: Wells Robert 
Published: July 2010

Let

f
:
M
→
M

f:M \to M

be a self-map of a closed smooth

n
n

-manifold. Does there exist a diffeomorphism

φ
:
M
→
M

\varphi :M \to M

homotopic to

f
f

? Define

φ
\varphi

to be almost regularly homotopic to the identity if

φ

|

M
−

p
t

\varphi |M - \mathrm {pt}

. is regularly homotopic to the inclusion

M
−

p
t

.
⊂
M

M - \mathrm {pt}. \subset M

. Let

ψ
:
M
→
M
∨
M

\psi :M \to M \vee M

be the result of collapsing the boundary of a smooth

n
n

-cell in

M
M

, and let

M
∨
M

→

Δ
′

M

M \vee M \stackrel {\Delta ’}{\to } M

be the codiagonal. For

ξ
∈

π
n

(
M
)

\xi \in {\pi _n}(M)

define

τ
(
ξ
)

\tau (\xi )

to be the composition
\[

M

→

ψ

M
∨
M

→

1
∨
ξ

M
∨
M

→

Δ
′

M
.

M \stackrel {\psi }{\rightarrow } M \vee M \stackrel {1 \vee \xi }{\rightarrow } M \vee M \stackrel {\Delta ’}{\rightarrow } M.

\]
Theorem. If

M
M

is

2
2

-connected,

s
s

-parallelizable, and

n
=
2
l
>
5

n = 2l > 5

with

l

≢

0
mod
(
4
)

l \nequiv 0 \bmod (4)

, then

τ
(
ξ
)

\tau (\xi )

contains a diffeomorphism almost regularly homotopic to the identity iff

ξ
\xi

is in the kernel of the stabilization map

π
n

(
M
)
→

π
n
s

(
M
)

{\pi _n}(M) \to \pi _n^s(M)

.
𝑅-separable coordinates for three-dimensional complex Riemannian spaces
Authors: Boyer C. P. , Kalnins E. G. , Miller Willard 
Published: July 2010

We classify allR-separable coordinate systems for the equationsΣi,j=13g−1/2∂j(g1/2gij∂iψ)=0\Sigma _{i,j = 1}^3\,{g^{ - 1/2}}{\partial _j}({g^{1/2}}{g^{ij}}{\partial _i}\psi ) = 0andΣi,j=13gij∂iW∂jW=0\Sigma _{i,j\, = \,1}^3 {{g^{ij}}{\partial _i}W{\partial _j}W\, = \,0}with special emphasis on nonorthogonal coordinates, and give a group-theoretic interpretation of the results. We show that for flat space the two equations separate in exactly the same coordinate systems.
Square integrable representations and a Plancherel theorem for parabolic subgroups
Authors: Keene Frederick W. 
Published: July 2010

Let G be a semisimple Lie group with Iwasawa decomposition

G

=

K
A
N

G\, = \,KAN

. Let

g

0

=

f

0

+

a

+

n

{\mathfrak {g}_0} \,=\, {\mathcal {f}_0} \,+\, \mathfrak {a} \,+\, \mathfrak {n}

be the corresponding decomposition of the Lie algebra of G. Then the nilpotent subgroup N has square integrable representations if and only if the reduced restricted root system is of type

A
1

{A_1}

or

A
2

{A_2}

. The Plancherel measure for N can be found explicitly in these cases. We then prove the Plancherel theorem in the

A
1

{A_1}

case for the solvable subgroup NA by combining Mackey’s “Little Group” method with an idea due to C. C. Moore: we find an operator D, defined on the

C
∞

{C^\infty }

functions on NA with compact support, such that
\[

ϕ
(
e
)
=

∫

(
N
A
)

∧

tr

(
D
π
(
ϕ
)
)
d
μ
(
π
)

\phi (e) = \int _{{{(NA)}^\wedge }} {{\text {tr}}} (D\pi (\phi ))d\mu (\pi )

\]
where

(
N
A

)
∧

{(NA)^ \wedge }

is the unitary dual, e is the identity, and

μ
\mu

is the Plancherel measure for NA, and D is an unbounded selfadjoint operator. In the

A
1

{A_1}

case, D involves fractional powers of the Laplace operator and hence is not a differential operator.
Branch point structure of covering maps onto nonorientable surfaces
Authors: Ezell Cloyd L. 
Published: July 2010

Let

f
:
M

→

N

f:M\, \to \,N

be a degree n branched cover onto a compact, connected nonorientable surface with branch points

y
1

,

y
2

,

…
,

y
m

{y_1},\,{y_2},\, \ldots ,\,{y_m}

in N, and let the multiplicities at points in

f

−
1

(

y
i

)

{f^{ - 1}}({y_i})

be

μ

i
1

,

μ

i
2

,

…
,

μ

i

k
i

{\mu _{i1}},\,{\mu _{i2}},\, \ldots ,\,{\mu _{i{k_i}}}

. The branching array of f, designated by B, is the following array of numbers:
\[

μ

11

,

μ

12

,

…
,

μ

1

k
1

μ

21

,

μ

22

,
…
,

μ

2

k
2

⋮

μ

m
1

,

μ

m
2

,

…
,

μ

m

k
m

\begin {gathered} {\mu _{11}},\,{\mu _{12}},\, \ldots ,\,{\mu _{1{k_1}}} {\mu _{21}},\,{\mu _{22\,}}, \ldots ,\,{\mu _{2{k_2}}} \,\,\,\,\,\,\,\,\,\,\,\, \vdots {\mu _{m1}},\,{\mu _{m2}},\, \ldots ,\,{\mu _{m{k_m}}} \end {gathered}

\]
We show that the numbers in the branching array must always satisfy the following conditions: (1)
\[

∑

{

μ

i
j

+

1

|

j

=

1
,

2
,

…
,

k
i

}

=

n

\sum {\{ {\mu _{ij}} \,+ \,1|j \,=\, 1,\,2,\, \ldots ,\,{k_i}\} \,=\, n}

\]
, (2)

∑

{

μ

i
j

|

i

=

1
,

2
,
…
,
m
;
j

=

1
,

2
,
…
,

k
i

}

\sum {\{ {\mu _{ij}}|i\,=\, 1,\,2, \ldots ,m;j\,=\, 1,\,2, \ldots ,{k_i}\} }

is even. Furthermore, if B is any array of numbers satisfying these conditions, and if N is not the projective plane, then there is a branched cover onto N with B as its branching array.
Sweedler’s two-cocycles and generalizations of theorems on Amitsur cohomology
Authors: Riffelmacher Dave 
Published: July 2010

For any (not necessarily commutative) algebra C over a commutative ring k Sweedler defined a cohomology set, denoted here by

H

2

(
C

/

k
)

{\mathcal {H}^2}(C/k)

, which generalizes Amitsur’s second cohomology group

H
2

(
C

/

k
)

{H^2}(C/k)

. In this paper, if I is a nilpotent ideal of C and

C
¯

≡

C

/

I

\bar C\, \equiv \,C/I

is K-projective, a natural bijection

H

2

(
C

/

k
)

→
~

H

2

(

C
¯

/

k
)

{\mathcal {H}^2}(C/k)\tilde \to {\mathcal {H}^2}(\bar C{\text {/}}k)

is established. Also, when

k
⊂
B

k \subset B

are fields and C is a commutative B-algebra, the sequence

{
1
}
→

H
2

(
B

/

k
)

→

l

∗

H
2

(
C

/

k
)

→

r

H
2

(
C

/

B
)

\{ 1\} \to {H^2}(B{\text {/}}k)\xrightarrow {{{l^{\ast }}}}{H^2}(C/k)\xrightarrow {r}{H^2}(C/B)

is shown to be exact if the natural map

C

⊗
k

C
→
C

⊗
B

C

C{ \otimes _k}C \to C{ \otimes _B}C

induces a surjection on units,

l

∗

{l^ {\ast } }

is induced by the inclusion, and r is the “restriction” map.
The Albanese mapping for a punctual Hilbert scheme. I. Irreducibility of the fibers
Authors: Huibregtse Mark E. 
Published: July 2010

Let

f
:

X
→
A

f:\,X \to A

be the canonical mapping from an algebraic surface X to its Albanese variety A,

X
(
n
)

X(n)

the n-fold symmetric product of X, and

H
X
n

H_X^n

the punctual Hilbert scheme parameterizing 0-dimensional closed subschemes of length n on X. The latter is a nonsingular and irreducible variety of dimension

2
n

2n

, and the “Hilbert-Chow” morphism

σ
n

:

H
X
n

→
X
(
n
)

{\sigma _n}:\,H_X^n \to X(n)

is a birational map which desingularizes

X
(
n
)

X(n)

. This paper studies the composite morphism
\[

φ
n

:

H
X
n

→

σ
n

X
(
n
)

→

f
n

A
,

{\varphi _n}:\,H_X^n\xrightarrow {{{\sigma _n}}}X(n)\xrightarrow {{{f_n}}}A ,

\]
where

f
n

{f_n}

is obtained from f by addition on A. The main result (Part 1 of the paper) is that for

n
≫
0

n \gg 0

, all the fibers of

φ
n

{\varphi _n}

are irreducible and of dimension

2
n
−
q

2n - q

, where

q
=
dim
⁡
A

q = \dim A

. An interesting special case (Part 2 of the paper) arises when

X
=
A

X = A

is an abelian surface; in this case we show (for example) that the fibers of

φ
n

{\varphi _n}

are nonsingular, provided n is prime to the characteristic.
Dispersion points for linear sets and approximate moduli for some stochastic processes
Authors: Geman Donald 
Published: July 2010

Let

Γ

∈

[
0
,

1
]

\Gamma \, \in \,[0,\,1]

be Lebesgue measurable; then

Γ
\Gamma

has Lebesgue density 0 at the origin if and only if
\[

∫
Γ

t

−
1

Ψ
(

t

−
1

meas

{
Γ

∩

(
0
,

t
)
}
)

d
t

>

∞

\int _\Gamma {{t^{ - 1}}\Psi ({t^{ - 1}}\,{\text {meas}}} \{ \Gamma \, \cap \,(0,\,t)\} )\,dt\, > \,\infty

\]
for some continuous, strictly increasing function

Ψ
(
t
)

(
0

⩽

t

⩽

1
)

\Psi (t)\,(0\, \leqslant \,t\, \leqslant \,1)

with

Ψ
(
0
)

=

0

\Psi (0)\, = \,0

. This result is applied to the local growth of certain Gaussian (and other) proceses

{

X
t

,

t

⩾

0
}

\{ {X_t},\,t\, \geqslant \,0\}

as follows: we find continuous, increasing functions

ϕ
(
t
)

\phi (t)

and

η
(
t
)

(
t

⩾

0
)

\eta (t)\,(t\, \geqslant \,0)

such that, with probability one, the set

{
t
:
η
(
t
)

⩽

|

X
t

−

X
0

|

⩽

ϕ
(
t
)
}

\{ t:\eta (t)\, \leqslant \,\left | {{X_t}\, - \,{X_0}} \right |\, \leqslant \,\phi (t)\}

has density 1 at the origin.
First Report of Anthracnose Caused by Colletotrichum gloeosporioides in Pumpkin in Trinidad
Authors: Rampersad S. N. 
Published: July 2010

In Trinidad, pumpkin (Cucurbita pepo L. and C. moschata L.) is extensively grown for local and international export markets. In November 2008, symptoms of foliar chlorosis and necrosis were observed in 15 commercial pumpkin fields located in the main production areas of St. George East, Caroni, Victoria, and St. Patrick counties. Severely infected plants were unable to support fruit maturation, which resulted in yield loss. The pathogen was isolated from surface-sterilized tissues of symptomatic plants. Colonies on potato dextrose agar (PDA) were white to cream with gray spore masses in the center. Conidia were hyaline, cylindrical with rounded ends, aseptate, and measured 12.5 to 16.5 μm × 3.5 to 5.0 μm. PCR amplification was carried out with ITS4/5 universal primers (4) and species-specific primers, CgInt/ITS4 (1), using a positive control of Colletotrichum gloeosporioides (courtesy of D. Perez-Brito). Species-specific primers generated a single amplicon, ~450 bp long, which corresponded with the positive control. The ITS1 region (1) of pumpkin isolates (GenBank No. GU320190) was 100% identical to cognate sequences of C. gloeosporioides isolates (GenBank Nos. AY841136 and FJ624257). Phylogenetic analyses (MEGA 4 – Molecular Evolutionary Genetic Analysis Software version 4 for Windows) using the neighbor-joining (NJ) algorithm placed the pumpkin isolates in a well-supported cluster (>90% bootstrap value based on 1,000 replicates) with other C. gloeosporioides isolates. The tree was rooted with C. crassipes (GenBank No. AJ536230). The pathogen was similar to C. gloeosporioides (Penz.) Penz. & Sacc. (3). In pathogenicity tests, six plants (cv. Jamaican squash) for each of five isolates were spray inoculated to runoff with a conidial suspension (1.0 × 106 conidia/ml). Negative controls were sprayed with sterile distilled water. In repeated tests, plants were symptomatic of infection 7 days postinoculation. There were no symptoms on control plants. Koch's postulates were fulfilled with the reisolation of the pathogen from symptomatic leaf tissues. Anthracnose is a serious threat to cucurbit production; however, infection is not common in pumpkin and squash (2). To my knowledge, this is the first report of C. gloeosporioides causing widespread anthracnose infection in pumpkin in Trinidad. References: (1) A. E. Brown et al. Phytopathology 86:523, 1996. (2) G. Kelly. Acta Hortic. (ISHS) 731:479, 2007. (3) B. C. Sutton. Page 1 in: Colletotrichum: Biology, Pathology and Control. CAB International. Wallingford, UK, 1992. (4) T. J. White et al. Page 315 in: PCR Protocols: A Guide to Methods and Applications. Academic Press, San Diego, 1990.