bluuebunny/crossref_metadata_2025_split huggingface dataset:
Local and global envelopes of holomorphy of domains in 𝐶²
Authors: Bedford Eric 
Published: June 2010

A criterion is given for a smoothly bounded domain

D
⊂

C

2

D \subset {{\mathbf {C}}^2}

to be locally extendible to a neighborhood of a point

z
0

∈
∂
D

{z_0} \in \partial D

. (This result may also be formulated in terms of extension of CR functions on

∂
D

\partial D

.) This is related to the envelope of holomorphy of the semitubular domain
\[

Ω
(
Φ
)
=
{
(
z
,
w
)
∈

C

2

:
Re
⁡
w
+

r
k

Φ
(
θ
)
>
0
}
,

\Omega (\Phi ) = \{ (z,w) \in {{\mathbf {C}}^2}:\operatorname {Re} w + {r^k}\Phi (\theta ) > 0\} ,

\]
where

r
=

|

z

|

r = |z|

,

θ
=
arg
⁡
(
z
)

\theta = \arg (z)

. Necessary and sufficient conditions are given for the envelope of holomorphy of

Ω
(
Φ
)

\Omega (\Phi )

to be

C

2

{{\mathbf {C}}^2}

. These conditions are equivalent to the existence of a subharmonic minorant for

r
k

Φ
(
θ
)

{r^k}\Phi (\theta )

.
Maximal functions associated with curves and the Calderón-Zygmund method of rotations
Authors: Sato Shuichi 
Published: June 2010

Let

δ
t

(
t
>
0
)

{\delta _t}(t > 0)

be a dilation in

R

n

(
n
⩾
2
)

{{\mathbf {R}}^n}(n \geqslant 2)

defined by
\[

δ
t

x
=
(

t

α
1

x
1

,

t

α
2

x
2

,
…
,

t

α
n

x
n

)

(
x
=
(

x
1

,

x
2

,
…
,

x
n

)
)
,

{\delta _t}x = ({t^{{\alpha _1}}}{x_1},{t^{{\alpha _2}}}{x_2}, \ldots ,{t^{{\alpha _n}}}{x_n})\qquad (x = ({x_1},{x_2}, \ldots ,{x_n})),

\]
where

α
i

>
0
(
i
=
1
,
2
,
…
,
n
)

{\alpha _i} > 0(i = 1,2, \ldots ,n)

and

α
i

≠

α
j

{\alpha _i} \ne {\alpha _j}

if

i
≠
j

i \ne j

. For

ν
∈

R

n

\nu \in {{\mathbf {R}}^n}

with

|

ν

|

=
1

|\nu | = 1

, let

Γ
ν

:
(
0
,
∞
)
→

R

n

{\Gamma _\nu }:(0,\infty ) \to {{\mathbf {R}}^n}

be a curve defined by

Γ
ν

(
t
)
=

δ
t

ν

(
0
>
t
>
∞
)

{\Gamma _\nu }(t) = {\delta _t}\nu \,(0 > t > \infty )

. Using maximal functions associated with the curves

Γ
ν

{\Gamma _\nu }

, we define an operator

M
M

which is a nonisotropic analogue of the one studied in R. Fefferman [2]. It is proved that

M
M

is a bounded operator on

L
p

(

R

n

)

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

for some

p
p

with

1
>
p
>
2

1 > p > 2

. As its application we prove the

L
p

{L^p}

boundedness of operators of the form

T

∗

(
f
)
(
x
)
=

sup

ε
>
0

|

T
ε

(
f
)
(
x
)

|

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

, where

T
ε

{T_\varepsilon }

is an integral operator associated with a variable kernel with mixed homogeneity.
Singularly perturbed quadratically nonlinear Dirichlet problems
Authors: DeSanti Albert J. 
Published: June 2010

The Dirichlet problem for singularly perturbed elliptic equations of the form

ε
Δ
u
=
A
(

x

,
u
)
∇
u
⋅
∇
u
+

B

(

x

,
u
)
⋅
∇
u
+
C
(

x

,
u
)

\varepsilon \Delta u = A({\mathbf {x}},u)\nabla u \cdot \nabla u + {\mathbf {B}}({\mathbf {x}},u) \cdot \nabla u + C({\mathbf {x}},u)

in

Ω
∈

E
n

\Omega \in {E^n}

is studied. Under explicit and easily checked conditions, solutions are shown to exist for

ε
\varepsilon

sufficiently small and to exhibit specified asymptotic behavior as

ε
→
0

\varepsilon \to 0

. The results are obtained using a method based on the theory of partial differential inequalities.
On the values at integers of the Dedekind zeta function of a real quadratic field
Authors: Kramer David 
Published: June 2010

In 1976 Shintani gave a decomposition of the Dedekind zeta function,

ζ
κ
(
s
)

\zeta \kappa (s)

, of a totally real number field into a finite sum of functions, each given by a Dirichlet series whose meromorphic continuation assumes rational values at negative integers. He obtained a formula for these values, thereby giving an expression for

ζ
κ
(
−
n
)
,

\zeta \kappa ( - n),

,

n
=
0
,

1
,

2
,
…

n = 0,\,1,\,2, \ldots

. Earlier, Zagier had studied the special case of

ζ
(
A
,

s
)

\zeta (A,\,s)

, the narrow ideal class zeta function for a real quadratic field. He decomposes

ζ
(
A
,

s
)

\zeta (A,\,s)

into

Σ
A

Z
Q

(
s
)

{\Sigma _A}{Z_Q}(s)

, where

Z
Q

(
s
)

{Z_Q}(s)

is given as a Dirichlet series associated to a binary quadratic form

Q
(
x
,

y
)
=
a

x
2

+
b
x
y
+
c

y
2

Q(x,\,y) = a{x^2} + bxy + c{y^2}

, and the summation is over a canonically given finite cycle of “reduced” quadratic forms associated to a narrow ideal class

A
A

. He then obtains a formula for

Z
Q

(
−
n
)

{Z_Q}( - n)

as a rational function in the coefficients of the form

Q
Q

. Since the denominator of

ζ
(
A
,

−
n
)

\zeta (A,\, - n)

is known not to depend on the class

A
A

, whereas the coefficients of reduced forms attain arbitrarily large values, it is natural to ask whether the rational function in Zagier’s formula might be replaced by a polynomial. In this paper such a result is obtained. For example, Zagier gives
\[

15120
ζ
(
A
,
−
2
)
=

∑
A

b
5

−
10
a

b
3

c
+
30

a
2

b

c
2

a
3

+

b
5

−
10
a

b
3

c
+
30

a
2

b

c
2

c
3

−
21
b
(
a
+
c
)

15120\zeta (A, - 2) = \sum \limits _A {\frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{a^3}}}} + \frac {{{b^5} - 10a{b^3}c + 30{a^2}b{c^2}}}{{{c^3}}} - 21b(a + c){\text { }}

\]
while our result is
\[

15120
ζ
(
A
,

−
2
)
=

1
2

(

∑
A

−

∑

θ
A

)

(
60

a
2

−
117
a
b
+
76
a
c
+
38

b
2

−
117
b
c
+
60

c
2

)

15120\zeta (A,\, - 2) = \frac {1} {2}\left ( {\sum \limits _A - \sum \limits _{\theta A} } \right )\,(60{a^2} - 117ab + 76ac + 38{b^2} - 117bc + 60{c^2})

\]
, where

θ
\theta

is the narrow ideal class consisting of principal ideals generated by elements of negative norm. Starting with a representation of

Z
Q

(
1
+
n
)

{Z_Q}(1 + n)

due to Shanks and Zagier for

n
=
1
,

2
,

3
,
…

n = 1,\,2,\,3, \ldots

as a certain transcendental function of the coefficients of

Q
Q

, we also obtain the result that

ζ
(
A
,

1
+
n
)

\zeta (A,\,1 + n)

is given as the same sum of reduced quadratic forms as in the formula for

ζ
(
A
,

−
n
)

\zeta (A,\, - n)

, times the appropriate “gamma factor.” This gives a new proof of the functional equation of

ζ
(
A
,

s
)

\zeta (A,\,s)

at integer values of

s
s

, and suggests the possibility that one might be able to prove the functional equation for all

s
s

by finding some relation between

Z
Q

(
s
)

{Z_Q}(s)

and

Z
Q

(
1
−
s
)

{Z_Q}(1 - s)

. So far we have not found such a relation.
𝐹-purity and rational singularity in graded complete intersection rings
Authors: Fedder Richard 
Published: June 2010

A simple criterion is given for determining “almost completely” whether the positively graded complete intersection ring

R
=
K
[

X
1

,

…
,

X

n
+
t

]

/

(

G
i

,

…
,

G
t

)

R = K[{X_1},\, \ldots ,\,{X_{n + t}}]/({G_i},\, \ldots ,\,{G_t})

, of dimension

n
n

, has an

F
F

-pure type singularity at

m
=
(

X
1

,

…
,

X

n
+
t

)

m = ({X_1},\, \ldots ,\,{X_{n + t}})

. Specifically, if

deg
⁡
(

X
i

)
=

α
i

>
0

\operatorname {deg} ({X_i}) = {\alpha _i} > 0

for

1
≤
i
≤
n
+
t

1 \leq i \leq n + t

and

deg
⁡
(

G
i

)
=

d
i

>
0

\operatorname {deg} ({G_i}) = {d_i} > 0

for

i
≤
i
≤
t

i \leq i \leq t

, then there exists an integer

δ
\delta

determined by the singular locus of

R
R

such that: (1)

R
R

has

F
F

-pure type if

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

>
δ

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > \delta

. (2)

R
R

does not have

F
F

-pure type if

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

>
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0

. The characterization given by this theorem is particularly effective if the singularity of

R
R

at

m
m

is isolated. In that case,

δ
=
0

\delta = 0

so that only the condition

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

=
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0

is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if

R
R

has an isolated rational singularity, then

R
R

has

F
F

-pure type. The converse is also “almost true” with the only exception being the case where

Σ

i
=
1

t

d
i

−

Σ

i
=
1

n
+
t

α
i

=
0

\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0

. In proving this criterion, a weak but more stable form of

F
F

-purity, called

F
F

-contractedness, is defined and explored.

R
R

is

F
F

-contracted (in characteristic

p
>
0

p > 0

) if every system of parameters for

m
m

is contracted with respect to the Frobenius map

F
:

R
→
R

F:\,R \to R

. Just as for

F
F

-purity, the notion of

F
F

-contracted type is defined in characteristic 0 by reduction to characteristic

p
p

. The two notions of

F
F

-pure (type) and

F
F

-contracted (type) coincide when

R
R

is Gorenstein; whence, in particular, when

R
R

is a complete intersection ring.
Positive solutions of systems of semilinear elliptic equations: the pendulum method
Authors: Glover Joseph 
Published: June 2010

Conditions are formulated which guarantee the existence of positive solutions for systems of the form
\[

−
Δ

u
1

+

f
1

(

u
1

,
…
,

u
n

)
=

μ
1

,

−
Δ

u
2

+

f
2

(

u
1

,
…
,

u
n

)
=

μ
2

,

⋮

⋮

⋮

⋮

−
Δ

u
n

+

f
n

(

u
1

,
…
,

u
n

)
=

μ
n

,

\begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots ,\,{u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \hfill \\ \end {gathered}

\]
, where

Δ
\Delta

is the Laplacian (with Dirichlet boundary conditions) on an open domain in

R

d

{\mathbf {R}^d}

, and where each

μ
i

{\mu _i}

is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where

∂

f
k

/

∂

x
j

>
0

\partial {f_k}/\partial {x_j} > 0

for every

k
k

and

j
j

.
Multilinear convolutions defined by measures on spheres
Authors: Oberlin Daniel M. 
Published: June 2010

Let

σ
\sigma

be Lebesgue measure on

Σ

n
−
1

{\Sigma _{n - 1}}

and write

σ
=
(

σ
1

,
…
,

σ
n

)

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

for an element of

Σ

n
−
1

{\Sigma _{n - 1}}

. For functions

f
1

,
…
,

f
n

{f_1}, \ldots ,{f_n}

on

R

{\mathbf {R}}

, define
\[

T
(

f
1

,
…
,

f
n

)
(
x
)
=

∫

Σ

n
−
1

f
1

(
x
−

σ
1

)
⋯

f
n

(
x
−

σ
n

)

d
σ
,

x
∈

R

.

T({f_1}, \ldots ,{f_n})(x) = \int _{{\Sigma _{n - 1}}} {{f_1}(x - {\sigma _1}) \cdots {f_n}(x - {\sigma _n})\,d\sigma ,\qquad x \in {\mathbf {R}}.}

\]
This paper partially answers the question: for which values of

p
p

and

q
q

is there an inequality
\[

|

|

T
(

f
1

,
…
,

f
n

)

|

|

q

⩽
C

|

|

f
1

|

|

p

⋯

|

|

f
n

|

|

p

?

||T({f_1}, \ldots ,{f_n})|{|_q} \leqslant C||{f_1}|{|_p} \cdots ||{f_n}|{|_p}?

\]
Generalized quotients in Coxeter groups
Authors: Björner Anders , Wachs Michelle L. 
Published: June 2010

For (

W
W

,

S
S

) a Coxeter group, we study sets of the form
\[

W

/

V
=
{
w
∈
W

|

l
(
w
v
)
=
l
(
w
)
+
l
(
v
)

for all

v
∈
V
}
,

W/V = \{ w \in W|l(wv) = l(w) + l(v)\;{\text {for all}}\;v \in V\} ,

\]
, where

V
⊆
W

V \subseteq W

. Such sets

W

/

V

W/V

, here called generalized quotients, are shown to have much of the rich combinatorial structure under Bruhat order that has previously been known only for the case when

V
⊆
S

V \subseteq S

(i.e., for minimal coset representatives modulo a parabolic subgroup). We show that Bruhat intervals in

W

/

V

W/V

, for general

V
⊆
W

V \subseteq W

, are lexicographically shellable. The Möbius function on

W

/

V

W/V

under Bruhat order takes values in

{
−
1
,

0
,

+
1
}

\{ - 1,\,0,\, + 1\}

. For finite groups

W
W

, generalized quotients are the same thing as lower intervals in the weak order. This is, however, in general not true. Connections with the weak order are explored and it is shown that

W

/

V

W/V

is always a complete meet-semilattice and a convex order ideal as a subset of

W
W

under weak order. Descent classes

D
I

=
{
w
∈
W

|

l
(
w
s
)
>
l
(
w
)
⇔
s
∈
I
,

for all

s
∈
S
}

{D_I} = \{ w \in W|l(ws) > l(w) \Leftrightarrow s \in I,\;{\text {for all}}\;s \in S\}

,

I
⊆
S

I \subseteq S

, are also analyzed using generalized quotients. It is shown that each descent class, as a poset under Bruhat order or weak order, is isomorphic to a generalized quotient under the corresponding ordering. The latter half of the paper is devoted to the symmetric group and to the study of some specific examples of generalized quotients which arise in combinatorics. For instance, the set of standard Young tableaux of a fixed shape or the set of linear extensions of a rooted forest, suitably interpreted, form generalized quotients. We prove a factorization result for the quotients that come from rooted forests, which shows that algebraically these quotients behave as a system of minimal "coset" representatives of a subset which is in general not a subgroup. We also study the rank generating function for certain quotients in the symmetric group.
On the complete 𝐺𝐿(𝑛,𝐶)-decomposition of the stable cohomology of 𝑔𝑙_{𝑛}(𝐴)
Authors: Hanlon Phil 
Published: June 2010

Let

A
A

be a graded, associative

C

{\mathbf {C}}

-algebra. For each

n
n

let

g

l
n

(
A
)

g{l_n}(A)

denote the Lie algebra of

n
×
n

n \times n

matrices with entries from

A
A

. In this paper we extend the Loday-Quillen theorem to nontrivial isotypic components of

G
L
(
n
,

C

)

GL(n,\,{\mathbf {C}})

acting on the Lie algebra cohomology of

g

l
n

(
A
)

g{l_n}(A)

. For

α
\alpha

and

β
\beta

partitions of some nonnegative integer

m
m

let

[
α
,

β

]
n

∈

Z

n

{[\alpha ,\,\beta ]_n} \in {{\mathbf {Z}}^n}

denote the maximal

G
L
(
n
,

C

)

GL(n,\,{\mathbf {C}})

-weight given by
\[

[
α
,

β

]
n

=

∑
i

α
i

e
i

−

∑
j

β
j

e

n
+
1
−
j

.

{[\alpha ,\,\beta ]_n} = \sum \limits _i {{\alpha _i}{e_i}} - \sum \limits _j {{\beta _j}{e_{n + 1 - j}}.}

\]
We show that the

[
α
,

β

]
n

{[\alpha ,\,\beta ]_n}

-isotypic component of the Lie algebra cohomology of

g

l
n

(
A
)

g{l_n}(A)

stabilizes when

n
→
∞

n \to \infty

and is equal to
\[

H
R

C

∗

(
A
)
⊗
(

H
~

∗

(
A
;

C

)

⊗
m

⊗

S
α

⊗

S
β

)

s
m

HR{C^{\ast }}(A) \otimes ({\tilde H^{\ast }}{(A;\,{\mathbf {C}})^{ \otimes m}} \otimes {S^\alpha } \otimes {S^\beta }){s_m}

\]
where

H
~

∗

(
A
;

C

)

{\tilde H^{\ast }}(A;\,{\mathbf {C}})

is the reduced Hochschild cohomology of

A
A

with trivial coefficients, where

H
R

C

∗

(
A
)

HR{C^{\ast }}(A)

is the graded exterior algebra generated by the cyclic cohomology of

A
A

, where

S
α

{S^\alpha }

and

S
β

{S^\beta }

are the irreducible

S
m

{S_m}

-modules indexed by

α
\alpha

and

β
\beta

and where the action of

S
m

{S_m}

on

H
~

(
A
;

C

)

⊗
m

\tilde H{(A;\,{\mathbf {C}})^{ \otimes m}}

is the exterior action.
Some inequalities for singular convolution operators in 𝐿^{𝑝}-spaces
Authors: Seeger Andreas 
Published: June 2010

Suppose that a bounded function

m
m

satisfies a localized multiplier condition

sup

t
>
0

|

|

ϕ
m
(

t
P

⋅
)

|

|

M
p

>
∞

{\sup _{t > 0}}||\phi m({t^P} \cdot )|{|_{{M_p}}} > \infty

, for some bump function

ϕ
\phi

. We show that under mild smoothness assumptions

m
m

is a Fourier multiplier in

L
p

{L^p}

. The approach uses the sharp maximal operator and Littlewood-Paley-theory. The method gives new results for lacunary maximal functions and for multipliers in Triebel-Lizorkin-spaces.
An Erdős-Wintner theorem for differences of additive functions
Authors: Hildebrand Adolf 
Published: June 2010

An Erdös-Wintner type criterion is given for the convergence of the distributions

D
x

(
z
)
=

[
x

]

−
1

#
{
1
⩽
n
⩽
x
:

f
(
n
+
1
)
−
f
(
n
)
⩽
z
}

{D_x}(z) = {[x]^{ - 1}}\# \{ 1 \leqslant n \leqslant x:\,f(n + 1) - f(n) \leqslant z\}

, where

f
f

is a real-valued additive function. A corollary of this result is that an additive function

f
f

, for which

f
(
n
+
1
)
−
f
(
n
)

f(n + 1) - f(n)

tends to zero on a set of density one, must be of the form

f
=
λ
log

f = \lambda \log

for some constant

λ
\lambda

. This had been conjectured by Erdős.
Cauchy integral equalities and applications
Authors: Choe Boo Rim 
Published: June 2010

We study bounded holomorphic functions

π
\pi

on the unit ball

B
n

{B_n}

of

C

n

{\mathbb {C}^n}

satisfying the following so-called Cauchy integral equalities:
\[

C
[

π

m
+
1

π
¯

]
=

γ
m

π
m

a
m
p
;

(
m
=
0
,
1
,
2
,
…
)

\begin {array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots )} \\ \end {array}

\]
for some sequence

γ
m

{\gamma _m}

depending on

π
\pi

. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on

B
n

{B_n}

, a projection theorem about the orthogonal projection of

H
2

(

B
n

)

{H^2}({B_n})

onto the closed subspace generated by holomorphic polynomials in

π
\pi

, and some new information about the inner functions. In particular, it is shown that if we interpret

BMOA

(

B
n

)

{\text {BMOA}}({B_n})

as the dual of

H
1

(

B
n

)

{H^1}({B_n})

, then the map

g
→
g
∘
π

g \to g \circ \pi

is a linear isometry of

BMOA

(

B
1

)

{\text {BMOA}}({B_1})

into

BMOA

(

B
n

)

{\text {BMOA}}({B_n})

for every inner function

π
\pi

on

B
n

{B_n}

such that

π
(
0
)
=
0

\pi (0) = 0

.
*-differential identities of prime rings with involution
Authors: Chuang Chen-Lian 
Published: June 2010

Main Theorem. Let

R
R

be a prime ring with involution

∗

^{\ast }

. Suppose that

ϕ
(

x
i

Δ
j

,

(

x
i

Δ
j

)

∗

)
=
0

\phi (x_i^{{\Delta _j}},{(x_i^{{\Delta _j}})^{\ast }}) = 0

is a

∗

{\ast }

-differential identity for

R
R

, where

Δ
j

{\Delta _j}

are distinct regular words of derivations in a basis

M
M

with respect to a linear order

>
>

on

M
M

. Then

ϕ
(

z

i
j

,

z

i
j

∗

)
=
0

\phi ({z_{ij}},z_{ij}^{\ast }) = 0

is a

∗

{\ast }

-generalized identity for

R
R

, where

z

i
j

{z_{ij}}

are distinct indeterminates. Along with the Main Theorem above, we also prove the following: Proposition 1. Suppose that

∗

^{\ast }

is of the second kind and that

C
C

is infinite. Then

R
R

is special. Proposition 2. Suppose that

S
W

(
V
)
⊆
R
⊆

L
W

(
V
)

{S_W}(V) \subseteq R \subseteq {L_W}(V)

. Then

Q
Q

, the two-sided quotient ring of

R
R

, is equal to

L
W

(
V
)

{L_W}(V)

. Proposition 3 (Density theorem). Suppose that

D

V

{}_DV

and

W
D

{W_D}

are dual spaces with respect to the nondegenerate bilinear form

(
,
)

(,)

. Let

v
1

,
…
,

v
s

,

v
s
′

,
…
,

v
s
′

∈
V

{v_1}, \ldots ,{v_s},\;v_s^\prime , \ldots ,v_s^\prime \in V

and

u
1

,
…
,

u
t

,

u
1
′

,
…
,

u
t
′

∈
W

{u_1}, \ldots ,{u_t},\;u_1^\prime , \ldots ,u_t^\prime \in W

be such that

{

v
1

,
…
,

v
s

}

\{ {v_1}, \ldots ,{v_s}\}

is

D
D

-independent in

V
V

and

{

u
1

,
…
,

u
t

}

\{ {u_1}, \ldots ,{u_t}\}

is

D
D

-independent in

W
W

. Then there exists

a
∈

S
W

(
V
)

a \in {S_W}(V)

such that

v
i

a
=

v
i
′

(
i
=
1
,
…
,
s
)

{v_i}a = v_i^\prime \,(i = 1, \ldots ,s)

and

a

∗

u
j

=

u
j
′

(
j
=
1
,
…
,
t
)

{a^{\ast }}{u_j} = u_j^\prime \,(j = 1, \ldots ,t)

if and only if

(

v
i
′

,

u
j

)
=
(

v
i

,

u
j
′

)

(v_i’,{u_j}) = ({v_i},u_j’)

for

i
=
1
,
…
,
s

i = 1, \ldots ,s

and

j
=
1
,
…
,
t

j = 1, \ldots ,t

. Proposition 4. Suppose that

R
R

is a prime ring with involution

∗

^{\ast }

and that

f
f

is a

∗

{\ast }

-generalized polynomial. If

f
f

vanishes on a nonzero ideal of

R
R

, than

f
f

vanishes on

Q
Q

, the two-sided quotient ring of

R
R

.
Global solvability of the derivative nonlinear Schrödinger equation
Authors: Lee Jyh-Hao 
Published: June 2010

The derivative nonlinear Schrödinger equation

(
DNLS
)

(\text {DNLS})

\[

i

q
t

=

q

x
x

±

(

q
∗

q
2

)

x

,

a
m
p
;

q
=
q
(
x
,
t
)
,
i
=

−
1

,

q
∗

(
z
)
=

q
(
z
)

¯

,

\begin {array}{*{20}{c}} {i{q_t} = {q_{xx}} \pm {{({q^\ast }{q^2})}_x},} & {q = q(x,t), i = \sqrt { - 1} ,{q^\ast }(z) = \overline {q(z)} ,} \\ \end {array}

\]
was first derived by plasma physicists [9,10]. This equation was used to interpret the propagation of circular polarized nonlinear Alfvén waves in plasma. Kaup and Newell obtained the soliton solutions of

DNLS
\text {DNLS}

in 1978 [5]. The author obtained the local solvability of

DNLS
\text {DNLS}

in his dissertation [6]. In this paper we obtain global existence (in time

t
t

) of Schwartz class solutions of

DNLS
\text {DNLS}

if the

L
2

{L^2}

-norm of the generic initial data

q
(
x
,
0
)

q(x,0)

is bounded.
On designs of maximal (+1,-1)-matrices of order 𝑛≡2(𝑚𝑜𝑑4). II
Authors: Yang C. H. 
Published: June 2010

Finding maximal

(
+
1
,
−
1
)

( + 1, - 1)

-matrices

M

2
m

{M_{2m}}

of order

2
m

2m

(with odd

m
m

) constructible in the standard form
\[

(

A

a
m
p
;
B

−

B
T

a
m
p
;

A
T

)

\left ( {\begin {array}{*{20}{c}} A & B \\ { - {B^T}} & {{A^T}} \\ \end {array} } \right )

\]
is reduced to the finding of two polynomials

C
(
w
)

C(w)

,

D
(
w
)

D(w)

(corresponding to the circulant submatrices

A
A

,

B
B

) satisfying

(

∗

)

|

C
(
w
)

|

2

+

|

D
(
w
)

|

2

=

1
2

(
m
−
1
)
,

\begin{equation}\tag {$*$} |C(w)|^2 + |D(w){|^2} = \tfrac {1}{2}(m - 1),\end{equation}

, where

w
w

is any primitive

m
m

th root of unity. Thus, all

M

2
m

{M_{2m}}

constructible by the standard form (see [4]) can be classified by the formula

(
∗
)

\left ( * \right )

. Some new matrices

M

2
m

{M_{2m}}

for

m
=
25
,
27
,
31

m = 25,27,31

, were found by this method.
A new type of Chebyshev quadrature
Authors: Barnhill R. E. , Dennis J. E. , Nielson G. M. 
Published: June 2010

A Chebyshev quadrature is of the form
\[

∫

−
1

1

w
(
x
)
f
(
x
)
d
x
≃

c

∑

k
=
1

n

f
(

x
k

)

\int _{ - 1}^1 {w(x)f(x)dx \simeq } c\sum \limits _{k = 1}^n {f({x_k})}

\]
It is usually desirable that the nodes

x
k

{x_k}

be in the interval of integration and that the quadrature be exact for as many monomials as possible (i.e., the first

n
+
1

n + 1

monomials). For

n
=
1
,
⋅
⋅
⋅
,
7

n = 1, \cdot \cdot \cdot ,7

and

9
9

, such a choice of nodes is possible, but for

n
=
8

n = 8

and

n
>
9

n > 9

, the nodes are complex. In this note, the idea used is that the

l
2

{l^2}

-norm of the deviations of the first

n
+
1

n + 1

monomials from their moments be a minimum. Numerical calculations are carried out for

n
=
8
,
10

n = 8,10

, and

11
11

and one interesting feature of the numerical results is that a “multiple” node at the origin is required. The above idea is then generalized to a minimization of the

l
2

{l^2}

-norm of the deviations of the first

k
k

monomials,

k
≧
n
+
1

k \geqq n + 1

, including

k
=
∞

k = \infty

, and corresponding numerical results are presented.
Stability of parabolic difference approximations to certain mixed initial boundary value problems
Authors: Osher Stanley 
Published: June 2010

We consider the equation
\[

u
t

−
a
(
x
,
t
)

u

x
x

−
b
(
x
,
t
)

u
x

−
c
(
x
,
t
)
u
=
f
(
x
,
t
)

{u_t} - a(x,t){u_{xx}} - b(x,t){u_x} - c(x,t)u = f(x,t)

\]
in a region

0
≦
x
≦
1
,
t
≧
0

0 \leqq x \leqq 1,t \geqq 0

, with inhomogeneous initial and boundary data. We are concerned with stability and estimates on divided differences in the maximum norm for solutions of consistent implicit, multistep, parabolic difference approximations to this problem. Using a parametrix approach, we give sufficient conditions for certain estimates to be valid.