bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the generators of the first homology with compact supports of the Weierstrass family in characteristic zero
Authors: Kato Goro C. 
Published: June 2010

Let

W

Q

=
Proj
⁡
(

Q

[

g

2

,

g
3

,
X
,
Y
,
Z
]

/

(

homogeneous ideal generated by

−

Y
2

Z
+
4

X
3

−

g
2

X

Z
2

−

g
3

Z
3

)
)

{{\mathbf {W}}_{\mathbf {Q}}} = \operatorname {Proj}({\mathbf {Q}}[{{\text {g}}_2},{g_3},X,Y,Z]/({\text {homogeneous ideal generated by }} - {Y^2}Z + 4\,{X^3} - {g_2}\,X{Z^2} - {g_3}\,{Z^3}))

. This is said to be the Weierstrass Family over the field

Q

{\mathbf {Q}}

. Then the first homology with compact supports of the Weierstrass Family is computed explicitly, i.e., it is generated by

{

C

−
k

d
X

∧

d
Y

}

k
⩾
1

{\{ {C^{ - k}}\,dX\, \wedge \;dY\}_{k \geqslant 1}}

and

{
X

C

−
k

d
X
∧

d
Y

}

k

⩾
1

{\{ X{C^{ - k}}dX \wedge \,dY\}_{k\, \geqslant 1}}

over the ring

Q

[

g

2

,

g
3

]

{\mathbf {Q}}[{g_{2}},{g_3}]

, where

C
C

is a polynomial

Y
2

−
4

X
3

+

g
2

X
+

g
3

{Y^2} - 4{X^3} + {g_2}X + {g_3}

. When one tensors the homology of the Weierstrass Family with

Δ

−
1

Q

[

g
2

,

g
3

]

{\Delta ^{ - 1}}\,{\mathbf {Q}}[{g_2},{g_3}]

, being localized at the discriminant

Δ
=

g
2
3

−
27

g
3
2

\Delta = g_2^3 - 27g_3^2

, over

Q

[

g

2

,

g
3

]

{\mathbf {Q}}[{{\text {g}}_2},{g_3}]

, the first homology is generated by

C

−
1

d
X

∧

d
Y

{C^{ - 1}}dX\; \wedge \;dY

and

X

C

−
1

d
X

∧
d
Y

X{C^{ - 1}}dX\, \wedge dY

. One also obtains the first homologies with compact supports of singular fibres over

℘
=
(

g
2

=

g

3

=
0
)

\wp = ({g_2} = {g_{3}} = 0)

and

℘
=

(

g
2

=
3
,

g

3

=
1
)

\wp = \,({g_2} = 3,{g_{3}} = 1)

as corollaries.
A linear homogenization problem with time dependent coefficient
Authors: Mascarenhas Maria Luisa 
Published: June 2010

We consider: the homogenization problem
\[

{

(
∂
u
ε

/

∂
t
)
(
x
,
t
)
+

β
ε

(
x
)

u
ε

(
x
,
t
)
=
0
,

a
m
p
;
t
⩽
0
,

u
ε

(
x
,
0
)
=
ϕ
(
x
)
,

\begin {cases} (\partial u\varepsilon /\partial t)(x,t) + \beta _\varepsilon (x) u_\varepsilon (x,t) = 0, & t\leqslant 0, \\ u_\varepsilon (x,0) = \phi (x), \end {cases}

\]
where

β
\beta

is a strictly positive bounded real function, periodic of period

1
1

, and

β
ε

(
x
)
=
β
(
x

/

ε
)

{\beta _\varepsilon }(x) = \beta (x/\varepsilon )

; the equivalent integral equation
\[

u
ε

(
x
,
t
)
+

∫
0
t

β
ε

(
x
)

u
ε

(
x
,
s
)

d
s
=
ϕ
(
x
)

;

{u_\varepsilon }(x,t) + \int _0^t {{\beta _\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)};

\]
and the homogenized equation
\[

u
0

(
x
,
t
)
+

∫
0
t

K
(
t
−
s
)

u
0

(
s
)

d
s
=
ϕ
(
x
)

,

{u_0}(x,t) + \int _0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)},

\]
where

K
K

is a unique, well-defined function depending on

β
\beta

. We study this problem for a time dependent

β
\beta

, and characterize a two-variable function

K
(
s
,
t
)

K(s,t)

satisfying
\[

u
0

(
x
,
t
)
+

∫
0
t

K
(
s
,
t
−
s
)

u
0

(
x
,
s
)

d
s
=
ϕ
(
x
)

{u_0}(x,t) + \int _0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)}

\]
and study its uniqueness.
James maps, Segal maps, and the Kahn-Priddy theorem
Authors: Caruso J. , Cohen F. R. , May J. P. , Taylor L. R. 
Published: June 2010

The standard combinatorial approximation

C
(

R
n

,
X
)

C({R^n},X)

to

Ω
n

Σ
n

X

{\Omega ^n}{\Sigma ^n}X

is a filtered space with easily understood filtration quotients

D
q

(

R
n

,
X
)

{D_q}({R^n},X)

. Stably,

C
(

R
n

,
X
)

C({R^n},X)

splits as the wedge of the

D
q

(

R
n

,
X
)

{D_q}({R^n},X)

. We here analyze the multiplicative properties of the James maps which give rise to the splitting and of various related combinatorially derived maps between iterated loop spaces. The target of the total James map

\[
j = (j_q): \Omega ^n \Sigma ^n X \bigtimes _{q \geqslant 0} \Omega ^{2nq} \Sigma ^{2nq} D_q(R^n, X) \]
is a ring space, and

j
j

is an exponential

H
H

-map. There is a total Segal map

\[
s = \bigtimes _{q \geqslant 0} \;{s_{q}}:\bigtimes _{q \geqslant 0} \;{\Omega ^{2nq}}\,{\Sigma ^{2nq}}{D_q}({R^{n}},X)\; \bigtimes _{q \geqslant 0} \;\Omega ^{3nq}\,\Sigma ^{3nq}{X^{[q]}}\]
which is a ring map between ring spaces. There is a total partial power map

\[
k = ({k_q}): {\Omega ^{n}}\,{\Sigma ^{n}}X \to \bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}}\]
which is an exponential

H
H

-map. There is a noncommutative binomial theorem for the computation of the smash power

Ω
n

Σ
n

X
→

Ω

n
q

Σ

n
q

X

[
q
]

{\Omega ^n}{\Sigma ^n}X \to {\Omega ^{nq}}{\Sigma ^{nq}}{X^{[q]}}

in terms of the

k
m

{k_m}

for

m
⩽
q

m \leqslant q

. The composite of

s
s

and

j
j

agrees with the composite of

k
k

and the natural inclusion

\[
\bigtimes _{q \geqslant 0} \;{\Omega ^{n\,q}}\,{\Sigma ^{n\,q}}{X^{[q]}} \to \bigtimes _{q \geqslant 0} \,{\Omega ^{3\,n\,q}}\,{\Sigma ^{3\,n\,q}}{X^{[q]}}.\]
This analysis applies to essentially arbitrary spaces

X
X

. When specialized to

X
=

S
0

X = {S^0}

, it implies an unstable version of the Kahn-Priddy theorem. The exponential property of the James maps leads to an analysis of the behavior of loop addition with respect to the stable splitting of

Ω
n

Σ
n

X

{\Omega ^n}{\Sigma ^n}X

when

X
X

is connected, and there is an analogous analysis relating loop addition to the stable splitting of

Q
(

X
+

)

Q({X^ + })

.
Localization of equivariant cohomology rings
Authors: Duflot J. 
Published: June 2010

The main result of this paper is the "calculation" of the Borel equivariant cohomology ring

H

∗

(
E
G

×
G

X
,

Z

/

p

Z

)

{H^{\ast } }(EG \times _G\,X,{\mathbf {Z}}/p{\mathbf {Z}})

localized at one of its minimal prime ideals. In case

X
X

is a point, the work of Quillen shows that the minimal primes

P

A

{\mathfrak {P}_A}

are parameterized by the maximal elementary abelian

p
p

-subgroups

A
A

of

G
G

and the result is
\[

H

∗

(
B
G
,

Z

/

p

Z

)

P

A

≅

H

∗

(
B

C
G

(
A
)
,

Z

/

p

Z

)

P

A

W
G

(
A
)

{H^{\ast } }{(BG,{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}} \cong {H^{\ast } }(B{C_G}(A),{\mathbf {Z}}/p{\mathbf {Z}})_{{\mathfrak {P}_A}}^{{W_G}(A)}

\]
. Here,

C
G

(
A
)

{C_G}(A)

is the centralizer of

A
A

in

G
G

, and

W
G

(
A
)
=

N
G

(
A
)

/

C
G

(
A
)

{W_G}(A) = {N_G}(A)/{C_G}(A)

, where

N
G

(
A
)

{N_G}(A)

is the normalizer of

A
A

in

G
G

. An example is included.
Extremal problems for polynomials with exponential weights
Authors: Mhaskar H. N. , Saff E. B. 
Published: June 2010

For the extremal problem:
\[

E

n
,
r

(
α
)
:=
min
∥
exp
⁡
(
−

|

x

|

α

)

(

x
n

+
⋯
)

∥

L
r

,

α
>
0
,

{E_{n,r}}(\alpha ): = \min \parallel \exp ( - |x{|^\alpha })\,({x^n} + \cdots ){\parallel _{{L^r}}}, \qquad \alpha > 0,

\]
where

L
r

(
0
>
r
⩽
∞
)

{L^r}\,(0 > r \leqslant \infty )

denotes the usual integral norm over

R

{\mathbf {R}}

, and the minimum is taken over all monic polynomials of degree

n
n

, we describe the asymptotic form of the error

E

n
,
r

(
α
)

(

as

n
→
∞
)

{E_{n,r}}(\alpha )\;({\text {as}}\;n \to \infty )

as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case

r
=
2

r = 2

yields new information regarding the polynomials

{

p
n

(
α
;
x
)
=

γ
n

(
α
)

x
n

+
⋯
}

\{ {p_n}(\alpha ;x) = {\gamma _n}(\alpha )\,{x^n} + \cdots \}

which are orthonormal on

R

{\mathbf {R}}

with respect to

exp
⁡
(
−
2

|

x

|

α

)

\exp ( - 2|x{|^\alpha })

. In particular, it is shown that a conjecture of Freud concerning the leading coefficients

γ
n

(
α
)

{\gamma _n}(\alpha )

is true in a Cesàro sense. Furthermore a contracted zero distribution theorem is proved which, unlike a previous result of Ullman, does not require the truth of the Freud’s conjecture. For

r
=
∞
,
α
>
0

r = \infty ,\alpha > 0

we also prove that, if

deg
⁡

P
n

(
x
)
⩽
n

\deg {P_n}(x) \leqslant n

, the norm

∥
exp
⁡
(
−

|

x

|

α

)

P
n

(
x
)

∥

L
∞

\parallel \exp ( - |x|^{\alpha })\,{P_n}(x)\parallel _{{L^\infty }}

is attained on the finite interval
\[

[

−

(
n

/

λ
α

)

1

/

α

,

(
n

/

λ
α

)

1

/

α

]

,

where

λ
α

=
Γ
(
α
)

/

2

α
−
2

{
Γ
(
α

/

2
)

}
2

.

\left [ { - {{(n/{\lambda _\alpha })}^{1/\alpha }},{{(n/{\lambda _\alpha })}^{1/\alpha }}} \right ],\quad {\text {where}}\;{\lambda _\alpha } = \Gamma (\alpha )/{2^{\alpha - 2}}{\{ \Gamma (\alpha /2)\} ^2}.

\]
Extensions of Nikolskii-type inequalities are also given.
Radial functions and invariant convolution operators
Authors: Meaney Christopher 
Published: June 2010

For

1
>
p
>
2

1 > p > 2

and

n
>
1

n > 1

, let

A
p

(

R

n

)

{A_p}({{\mathbf {R}}^n})

denote the Figà-Talamanca-Herz algebra, consisting of functions of the form

(
∗
)

( \ast )

\[

∑

k
=
0

∞

f
k

∗

g
k

\sum \limits _{k = 0}^\infty {{f_k} \ast {g_k}}

\]
with

∑
k

|

|

f
k

|

|

p

⋅

|

|

g
k

|

|

p
′

>
∞

\sum \nolimits _k {||{f_k}|{|_p}\cdot ||{g_k}|{|_{p\prime }} > \infty }

. We show that if

2
n

/

(
n
+
1
)
>
p
>
2

2n/(n + 1) > p > 2

, then the subalgebra of radial functions in

A
p

(

R

n

)

{A_p}({{\mathbf {R}}^n})

is strictly larger than the subspace of functions with expansions

(
∗
)

( \ast )

subject to the additional condition that

f
k

{f_k}

and

g
k

{g_k}

are radial for all

k
k

. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix

G
∈
[
F
I
A

]

B

−

G \in [FIA]_{B}^{ - }

and consider

B

A
p

(
G
)

^B{A_p}(G)

the subalgebra of

B
B

-invariant elements of

A
p

(
G
)

{A_p}(G)

. In particular, we show that the dual of

B

A
p

(
G
)

^B{A_p}(G)

is equal to the space of bounded, right-translation invariant operators on

L

p

(
G
)

{L^{p}}(G)

which commute with the action of

B
B

.
The normal closure of the coproduct of rings over a division ring
Authors: Martindale Wallace S. 
Published: June 2010

Let

R
=

R
1

∐

R
2

R = {R_1}\coprod {R_2}

be the coproduct of

Δ
\Delta

-rings

R
1

{R_1}

and

R
2

{R_2}

with 1 over a division ring

Δ
,

R
1

≠
Δ
,

R
2

≠
Δ

\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta

, with at least one of the dimensions

(

R
i

:
Δ

)
r

,

(

R
i

:
Δ

)
l

,

i
=
1
,

2

{({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2

, greater than 2. If

R
1

{R_1}

and

R
2

{R_2}

are weakly

1
1

-finite (i.e., one-sided inverses are two-sided) then it is shown that every

X
X

-inner automorphism of

R
R

(in the sense of Kharchenko) is inner, unless

R
1

,

R
2

{R_1},\,{R_2}

satisfy one of the following conditions: (I) each

R
i

{R_i}

is primary (i.e.,

R
i

=
Δ
+
T
,

T
2

=
0

{R_i} = \Delta + T,\,{T^2} = 0

), (II) one

R
i

{R_i}

is primary and the other is

2
2

-dimensional, (III) char.

Δ
=
2

\Delta = 2

, one

R
i

{R_i}

is not a domain, and one

R
i

{R_i}

is

2
2

-dimensional. This generalizes a recent joint result with Lichtman (where each

R
i

{R_i}

was a domain) and an earlier joint result with Montgomery (where each

R
i

{R_i}

was a domain and

Δ
\Delta

was a field).
Global boundedness for a delay-differential equation
Authors: Luckhaus Stephan 
Published: June 2010

The inequality

(

∂
t

u
−
Δ
u
)
(
t
,

x
)

≤

u
(
t
,

x
)
(
1
−
u
(
t
−
τ
,

x
)
)

({\partial _t}u - \Delta u)(t,\,x)\qquad \leq \qquad u(t,\,x)(1 - u(t - \tau ,\,x))

is investigated. It is shown that nonnegative solutions of the Dirichlet problem in a bounded interval remain bounded as time goes to infinity, whereas in a more dimensional domain, in general, this holds only if the delay is not too large.
Inequalities for some maximal functions. II
Authors: Cowling M. , Mauceri G. 
Published: June 2010

Let

S
S

be a smooth compact hypersurface in

R

n

{{\mathbf {R}}^n}

, and let

μ
\mu

be a measure on

S
S

, absolutely continuous with respect to surface measure. For

t
t

in

R

+

,

μ
t

{{\mathbf {R}}^ + },{\mu _t}

denotes the dilate of

μ
\mu

by

t
t

, normalised to have the same total variation as

μ
\mu

: for

f
f

in

S

(

R

n

)
,

μ
#

f

\mathcal {S}({{\mathbf {R}}^n}),{\mu ^\# }f

denotes the maximal function

sup

t
>
0

|

μ
t

∗
f

|

{\sup _{t > 0}}|{\mu _t}\ast f|

. We seek conditions on

μ
\mu

which guarantee that the a priori estimate
\[

‖

μ
#

f
‖

p

≤
C

‖
f
‖

p

,

f
∈
S
(

R

n

)
,

\left \| \mu ^\# f\right \|_p \leq C\left \| f \right \|_p, \quad f \in S(\mathbf {R}^n),

\]
holds; this estimate entails that the sublinear operator

μ
#

{\mu ^\# }

extends to a bounded operator on the Lebesgue space

L
p

(

R

n

)

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

. Our methods generalise E. M. Stein’s treatment of the "spherical maximal function" [5]: a study of "Riesz operators",

g
g

-functions, and analytic families of measures reduces the problem to that of obtaining decay estimates for the Fourier transform of

μ
\mu

. These depend on the geometry of

S
S

and the relation between

μ
\mu

and surface measure on

S
S

. In particular, we find that there are natural geometric maximal operators limited on

L
p

(

R

n

)

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

if and only if

p
∈
(
q
,
∞
]
;
q

p \in (q,\infty ];q

is some number in

(
1
,
∞
)

(1,\infty )

, and may be greater than

2
2

. This answers a question of S. Wainger posed by Stein [6]>.
Endomorphisms of right ideals of the Weyl algebra
Authors: Stafford J. T. 
Published: June 2010

Let

A
=
A
(
k
)

A = A(k)

be the first Weyl algebra over an infinite field

k
k

, let

P
P

be any noncyclic, projective right ideal of

A
A

and set

S
=
End
⁡
(
P
)

S = \operatorname {End} (P)

. We prove that, as

k
k

-algebras,

S
≇
A

S\not \cong A

. In contrast, there exists a noncyclic, projective right ideal

Q
Q

of

S
S

such that

S
≅
End
⁡
(
Q
)

S \cong \operatorname {End} (Q)

. Thus, despite the fact that they are Morita equivalent,

S
S

and

A
A

have surprisingly different properties. For example, under the canonical maps,

Aut
k

(
A
)
≅

Pic
k

(
A
)
≅

Pic
k

(
S
)

{\operatorname {Aut} _k}(A) \cong {\operatorname {Pic} _k}(A) \cong {\operatorname {Pic} _k}(S)

. In contrast,

Aut
k

(
S
)

{\operatorname {Aut} _k}(S)

has infinite index in

Pic
k

(
S
)

{\operatorname {Pic} _k}(S)

.
On inductive limits of matrix algebras of holomorphic functions
Authors: Peters Justin 
Published: June 2010

Let

A

\mathfrak {A}

be a UHF algebra and

A

(

D

)

\mathcal {A}({\mathbf {D}})

the disk algebra. If

A

=

[

∪

n
≥
1

A

n

]

−

\mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - }

and

α
\alpha

is a product-type automorphism of

A

\mathfrak {A}

which leaves each

A

n

{\mathfrak {A}_n}

invariant, then

α
\alpha

defines an embedding
\[

A

n

⊗

A

(

D

)

↪

ı
n

A

n
+
1

⊗

A

(

D

)

\mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}})

\]
. The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum

Γ
(
α
)

\Gamma (\alpha )

is finite.
Best constants in norm inequalities for the difference operator
Authors: Kaper Hans G. , Spellman Beth E. 
Published: June 2010

Let

ξ
=

(

ξ
m

)

m
∈

Z

\xi = {({\xi _m})_{m \in {\mathbf {Z}}}}

be an arbitrary element of the sequence space

l
∞

(

Z

)

{l^\infty }({\mathbf {Z}})

, and let

Δ
\Delta

be the difference operator on

l
∞

(

Z

)
:
Δ
ξ
=

(

ξ

m
+
1

−

ξ
m

)

m
∈

Z

{l^\infty }({\mathbf {Z}}):\Delta \xi = {({\xi _{m + 1}} - {\xi _m})_{m \in {\mathbf {Z}}}}

. The object of this investigation is the best possible value
\[

C
(
n
,

k
)
=

sup

{

Q

n
,
k

(
ξ
)
:
ξ
∈

l
∞

(

Z

)
,

Δ
n

ξ
≠
0
}

C(n,\,k) = {\operatorname {sup}}\{ {Q_{n,k}}(\xi ):\xi \in {l^\infty }({\mathbf {Z}}),\,{\Delta ^n}\xi \ne 0\}

\]
of the quotient
\[

Q

n
,
k

(
ξ
)
=

‖

Δ
k

ξ

‖

‖
ξ
‖

(
n
−
k
)

/

n

‖

Δ
n

ξ

‖

k

/

n

{Q_{n,k}}(\xi ) = \frac {{\left \| {{\Delta ^k}\xi } \right \|}} {{{{\left \| \xi \right \|}^{(n - k)/n}}{{\left \| {{\Delta ^n}\xi } \right \|}^{k/n}}}}

\]
, where

n
=
2
,

3
,
…

n = 2,\:3, \ldots

;

k
=
1
,
…
,

n
−
1

k = 1, \ldots ,\,n - 1

. It is shown that

C
(
n
,

k
)

C(n,\,k)

is at least equal to the corresponding constant

K
(
n
,

k
)

K(n,\,k)

, determined by Kolmogorov [Moscov. Gos. Univ. Uchen. Zap. Mat. 30 (1939), 3-13; Amer. Math. Soc. Transl. (1) 2 (1962), 233-243] for the differential operator

D
D

on

L
∞

(

R

)

{L^\infty }({\mathbf {R}})

, and exactly equal to

K
(
n
,

k
)

K(n,\,k)

if

k
=
n
−
1

k = n - 1

. Lower bounds for

C
(
n
,

k
)

C(n,\,k)

are derived that show that

C
(
n
,

k
)

C(n,\,k)

is generally greater than

K
(
n
,

k
)

K(n,\,k)

. The values of

C
(
n
,

k
)

C(n,\,k)

,

k
=
1
,
…
,

n
−
1

k = 1, \ldots ,\,n - 1

, are computed for

n
=
2
,
…
,
5

n = 2, \ldots ,5

.
On the central limit theorem for dynamical systems
Authors: Burton Robert , Denker Manfred 
Published: June 2010

Given an aperiodic dynamical system

(
X
,
T
,
μ
)

(X,T,\mu )

then there is an

f
∈

L
2

(
μ
)

f \in {L^2}(\mu )

with

∫
f
d
μ
=
0

\smallint fd\mu = 0

satisfying the Central Limit Theorem, i.e. if

S
m

f
=
f
+
f
∘
T
+
⋯
+
f
∘

T

m
−
1

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

and

σ
m

=

‖

S
m

f

‖

2

{\sigma _m} = {\left \| {{S_m}f} \right \|_2}

then
\[

μ

{

x

|

S
m

f
(
x
)

σ
m

>
u

}

→

(
2
π

)

−
1

/

2

∫

−
∞

u

exp

[

−

υ
2

2

]

d
υ
.

\mu \left \{ {x|\frac {{{S_m}f(x)}}{{{\sigma _m}}} > u} \right \} \to {(2\pi )^{ - 1/2}}\int _{ - \infty }^u {{\text {exp}}} \left [ {\frac {{ - {\upsilon ^2}}}{2}} \right ]d\upsilon .

\]
The analogous result also holds for flows.
On the generalized spectrum for second-order elliptic systems
Authors: Cantrell Robert Stephen , Cosner Chris 
Published: June 2010

We consider the system of homogeneous Dirichlet boundary value problems

(

∗

)

({\ast })

\[

L
1

u
=
λ
[

a

11

(
x
)
u
+

a

12

(
x
)
v
]
,

L
2

v
=
μ
[

a

12

(
x
)
u
+

a

22

(
x
)
v
]

{L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]

\]
in a smooth bounded domain

Ω
⊆

R

N

\Omega \subseteq {{\mathbf {R}}^N}

, where

L
1

{L_1}

and

L
2

{L_2}

are formally self-adjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of

(

∗

)

({\ast })

, i.e., the set

(
λ
,

μ
)
∈

R

2

:

(

∗

)

(\lambda ,\,\mu ) \in {{\mathbf {R}}^2}:\,({\ast })

has a nontrivial solution. The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
Rings of differential operators on invariant rings of tori
Authors: Musson Ian M. 
Published: June 2010

Let

k
k

be an algebraically closed field of characteristic zero and

G
G

a torus acting diagonally on

k
s

{k^s}

. For a subset

β
\beta

of

s

=
{
1
,

2
,
…
,

s
}

{\mathbf {s}} = \{ 1,\,2, \ldots ,\,s\}

, set

U
β

=
{
u
∈

k
s

|

u
j

≠
0

if

j
∈
β
}

{U_\beta } = \{ u \in {k^s}|{u_j} \ne 0\;{\text {if}}\;j \in \beta \}

. Then

G
G

acts on

O

(

U
β

)

\mathcal {O}({U_\beta })

, the ring of regular functions on

U
β

{U_\beta }

, and we study the ring

D
(

O

(

U
β

)
G

)

D(\mathcal {O}{({U_\beta })^G})

of all differential operators on the invariant ring. More generally suppose that

Δ
\Delta

is a set of subsets of s, such that each invariant ring

O

(

U
β

)
G

\mathcal {O}{({U_\beta })^G}

,

β
∈
Δ

\beta \in \Delta

, has the same quotient field. We prove that

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

is Noetherian and finitely generated as a

k
k

-algebra. Now

G
G

acts on each

D
(

O

(

U
β

)
)

D(\mathcal {O}({U_\beta }))

and there is a natural map
\[

θ
:

⋂

β
∈
Δ

D

(

O

(

U
β

)
)

G

→

⋂

β
∈
Δ

D
(

O

(

U
β

)

G

)
=
D
(

Y
Δ

/

G
)

\theta :\bigcap \limits _{\beta \in \Delta } {D{{(\mathcal {O}({U_\beta }))}^G} \to \bigcap \limits _{\beta \in \Delta } {D(\mathcal {O}{{({U_\beta })}^G}) = D({Y_\Delta } / G)} }

\]
obtained by restriction of the differential operators. We find necessary and sufficient conditions for

θ
\theta

to be surjective and describe the kernel of

θ
\theta

. The algebras

∩

β
∈
Δ

D

(

O

(

U
β

)

)
G

{ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}

and

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

carry a natural filtration given by the order of the differential operators. We show that the associated graded rings are finitely generated commutative algebras and are Gorensetin rings. We also determine the centers of

∩

β
∈
Δ

D

(

O

(

U
β

)

)
G

{ \cap _{\beta \in \Delta }}D{(\mathcal {O}({U_\beta }))^G}

and

∩

β
∈
Δ

D
(

O

(

U
β

)
G

)

{ \cap _{\beta \in \Delta }}D(\mathcal {O}{({U_\beta })^G})

.
Local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations
Authors: Lieberman Gary M. 
Published: June 2010

We consider solutions (and subsolutions and supersolutions) of the boundary value problem
\[

a

i
j

(
x
,

u
,

D
u
)

D

i
j

u
+
a
(
x
,

u
,

D
u
)
=
0

in

Ω
,

β
i

(
x
)

D
i

u
+
γ
(
x
)
u
=
g
(
x
)

on

∂
Ω

\begin {array}{*{20}{c}} {{a^{ij}}(x,\,u,\,Du){D_{ij}}u + a(x,\,u,\,Du) = 0\quad {\text {in}}\;\Omega ,} \\ {{\beta ^i}(x){D_i}u + \gamma (x)u = g(x)\quad {\text {on}}\;\partial \Omega } \\ \end {array}

\]
for a Lipschitz domain

Ω
\Omega

, a positive-definite matrix-valued function

[

a

i
j

]

[{a^{ij}}]

, and a vector field

β
\beta

which points uniformly into

Ω
\Omega

. Without making any continuity assumptions on the known functions, we prove Harnack and Hölder estimates for

u
u

near

∂
Ω

\partial \Omega

. In addition we bound the

L
∞

{L^\infty }

norm of

u
u

near

∂
Ω

\partial \Omega

in terms of an appropriate

L
p

{L^p}

norm and the known functions. Our approach is based on that for the corresponding interior estimates of Trudinger.
Linear series with cusps and 𝑛-fold points
Authors: Schubert David 
Published: June 2010

A linear series

(
V
,

L

)

(V,\,\mathcal {L})

on a curve

X
X

has an

n
n

-fold point along a divisor

D
D

of degree

n
n

if

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
D
)
)
)
⩾
dim
⁡
(
V
)
−
1

\dim (V \cap {H^0}(X,\,\mathcal {L}( - D))) \geqslant \dim (V) - 1

. The linear series has a cusp of order

e
e

at a point

P
P

if

dim
⁡
(
V
∩

H
0

(
X
,

L

(
−
(
e
+
1
)
P
)
)
)
⩾
dim
⁡
(
V
)
−
1

\dim (V \cap {H^0}(X,\,\mathcal {L}( - (e + 1)P))) \geqslant \dim (V) - 1

. Linear series with cusps and

n
n

-fold points are shown to exist if certain inequalities are satisfied. The dimensions of the families of linear series with cusps are determined for general curves.
Infinitely many periodic solutions for the equation: 𝑢_{𝑡𝑡}-𝑢ₓₓ±|𝑢|^{𝑝-1}𝑢=𝑓(𝑥,𝑡). II
Authors: Tanaka Kazunaga 
Published: June 2010

Existence of forced vibrations of nonlinear wave equation:
\[

u

t
t

−

u

x
x

±

|

u

|

p
−
1

u
=
f
(
x
,

t
)
,

a
m
p
;

(
x
,

t
)
∈
(
0
,

π
)
×

R

,

u
(
0
,

t
)
=
u
(
π
,

t
)
=
0
,

a
m
p
;

t
∈

R

,

u
(
x
,

t
+
2
π
)
=
u
(
x
,

t
)
,

a
m
p
;

(
x
,

t
)
∈
(
0
,

π
)
×

R

,

\begin {array}{*{20}{c}} {{u_{tt}} - {u_{xx}} \pm |u{|^{p - 1}}u = f(x,\,t),} \hfill & {(x,\,t) \in (0,\,\pi ) \times {\mathbf {R}},} \hfill \\ {u(0,\,t) = u(\pi ,\,t) = 0,} \hfill & {t \in {\mathbf {R}},} \hfill \\ {u(x,\,t + 2\pi ) = u(x,\,t),} \hfill & {(x,\,t) \in (0,\,\pi ) \times {\mathbf {R}},} \hfill \\ \end {array}

\]
is considered. For all

p
∈
(
1
,

∞
)

p \in (1,\,\infty )

and

f
(
x
,

t
)
∈

L

(
p
+
1
)

/

p

f(x,\,t) \in {L^{(p + 1)/p}}

, existence of infinitely many periodic solutions is proved. This improves the results of the author [29, 30]. We use variational methods to show the existence result. Minimax arguments and energy estimates for the corresponding functional play an essential role in the proof.