bluuebunny/crossref_metadata_2025_split huggingface dataset:
Gauss sums and Fourier analysis on multiplicative subgroups of 𝑍_{𝑞}
Authors: Diamond Harold G. , Gerth Frank , Vaaler Jeffrey D. 
Published: June 2010

LetG(q)G(q)denote the multiplicative group of invertible elements inZq{{\mathbf {Z}}_q}, the ring of integers moduloqq. LetH⊆G(q)H \subseteq G(q)be a multiplicative subgroup with cosetsaHaHandbHbH. Iff: Zq→Cf:\ {\mathbf {Z}}_q \to {\mathbf {C}}is supported inaHaHwe show thatffcan be recovered from the values off^\hat frestricted tobHbHif and only if Gauss sums forHHare nonvanishing. Heref^\hat fis the (finite) Fourier transform offfwith respect to the additive groupZq{{\mathbf {Z}}_q}. The main result is a simple criterion for deciding when these Gauss sums are nonvanishing. IfH=G(q)H = G(q)thenffcan be recovered fromf^\hat frestricted toG(q)G(q)by a particularly elementary formula. This formula provides some inequalities and extremal functions.
The sufficiency of the Matkowsky condition in the problem of resonance
Authors: Lin Ching Her 
Published: June 2010

We consider the sufficiency of the Matkowsky condition concerning the differential equation

ε
y
+
f
(
x
,
ε
)

y
′

+
g
(
x
,
ε
)
y
=
0

(
−
a
⩽
x
⩽
b
)

\varepsilon y + f(x,\varepsilon )y’ + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)

under the assumption that

f
(
0
,
ε
)
=
0

f(0,\varepsilon ) = 0

identically in

ε
,

f
x

(
0
,
ε
)
≠
0

\varepsilon ,{f_x}(0,\varepsilon ) \ne 0

with

f
>
0

f > 0

for

x
>
0

x > 0

and

f
>
0

f > 0

for

x
>
0

x > 0

. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if

f
f

and

g
g

are convergent power series for

|

ε

|

>
ρ

(
ρ
>
0
)
,
f
(
x
,
0
)
=
−
2
x

|\varepsilon | > \rho \;(\rho > 0),f(x,0)=-2x

and the interval

[
−
a
,
b
]

[ - a,b]

is contained in a disc

D
D

with center at

0
0

. The main problem in this work is to remove from Sibuya’s result the assumption that

D
D

is a disc.
Geometric condition for universal interpolation in ℰ̂’
Authors: Squires William A. 
Published: June 2010

It is known that if

h
h

is an entire function of exponential type and

Z
(
h
)
=

{

z
k

}

k
=
1

Z(h) = {\{ {z_k}\} _{k = 1}}

with

|

h
′

(

z
k

)

|

⩾
ε
exp
⁡
(
−
c

|

z
k

|

)

|h’({z_k})| \geqslant \varepsilon \exp (- c|{z_k}|)

for constants

ϵ
\epsilon

,

C
C

independent of

k
k

, then

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

is a universal interpolation sequence. That is, given any sequence of complex numbers

{

a
k

}

k
=
1

∞

\{ {a_k}\} _{k = 1}^\infty

such that

|

a
k

|

⩽
A

exp
⁡
(
B

|

z
k

|

)

|{a_k}| \leqslant A\,\exp (B|{z_k}|)

for constants

A
,
B

A,B

independent of

K
K

then there exists

g
g

of exponential type such that

g
(

z
k

)
=

a
k

g({z_k}) = {a_k}

. This note is concerned with finding geometric conditions which make

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

a universal interpolation sequence for various spaces of entire functions. For the space of entire functions of exponential type a necessary and sufficient condition for

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

to be a universal interpolation sequence is that

∫
0

|

z
k

|

n
(

z
k

,
t
)

d
t

/

t
⩽
C

|

z
k

|

+
D
,
k
=
1

,
2
,
…

\int _0^{|{z_k}|} {n({z_k},t)\,dt/t \leqslant C|{z_k}| + D,k = 1} , 2,\ldots

, where

n
(

z
k

,
t
)

n({z_k},t)

is the number of points of

{

z
k

}

k
=
1

∞

\{ {z_k}\} _{k = 1}^\infty

in the disc of radius

t
t

about

z
k

{z_k}

, excluding

z
k

{z_k}

, and

C
,
D

C,D

are constants independent of

k
k

. Results for the space

E

^

′

=
{
f

entire

|

|

f
(
z
)

|

⩽
A

exp
⁡
[
B

|

Im
⁡
z

|

+
B
log
⁡
(
1
+

|

z

|

2

)
]
}

\hat {\mathcal {E}}^\prime = \{ f\;{\text {entire}}||f(z)| \leqslant A\;\exp [B|\operatorname {Im} z| + B\log (1 + |z|^{2})]\}

are given but the theory is not as complete as for the above example.
Szegő limit theorems for the harmonic oscillator
Authors: Janssen A. J. E. M. , Zelditch Steven 
Published: June 2010

Let

H
=
−

1
2

d
2

/

d

x
2

+

1
2

x
2

H = - \frac {1}{2}{d^2}/d{x^2} + \frac {1}{2}{x^2}

be the harmonic oscillator Hamiltonian on

L
2

(

R

)

{L^2}( {\mathbf {R}})

, and let

A
A

be a selfadjoint

D
O

DO

of order

O
O

in the Beals-Fefferman class with weights

φ
=
1
,
Φ
(
x
,
ξ
)
=

(
1
+

|

ξ

|

2

+

|

x

|

2

)

1

/

2

\varphi = 1,\Phi (x,\xi ) = {(1 + |\xi \,{|^2} + |x\,{|^2})^{1/2}}

. Form the measure

μ
(
f
)
=

lim

λ
→
∞

(
1

/

rank

π
λ

)

tr

f
(

π
λ

A

π
λ

)

\mu (f) = {\lim _{\lambda \to \infty }}(1/{\text {rank}}\;{\pi _\lambda })\,{\text {tr}}\,f({\pi _\lambda }\,A{\pi _\lambda })

where

π
λ

A

π
λ

{\pi _\lambda }\,A{\pi _\lambda }

is the compression of

A
A

onto the span of the Hermite functions with eigenvalue less than or equal to

λ
\lambda

. Then one has the following Szegö limit theorem:
\[

μ
(
f
)
=

lim

T
→
∞

1

2

π

T

∫

H
(
x
,
ξ
)
⩽
T

f
(
a
(
x
,
ξ
)
)

d
x

d
ξ

for

f
∈
C
(

R

)
.

\mu (f) = \lim \limits _{T \to \infty } \;\frac {1} {{2\,\pi \,T}}\;\int _{H(x,\xi ) \leqslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text {for}}\ f \in C({\mathbf {R}}).

\]
For the special case where

f
(
x
)
=
x

f(x) = x

, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.
Length dependence of solutions of FitzHugh-Nagumo equations
Authors: Collins Clyde 
Published: June 2010

We investigate the behavior of the solutions of the problem
\[

u
t

=

u

x
x

−
α
u
−
υ
+

u
2

(
1
+
α
−
u
)
,

a
m
p
;

υ
t

=
η

υ

x
x

+
σ
u
−
γ
υ
,

u
(
0
,
t
)
=
g
(
t
)
,

υ
(
0
,
t
)
=
h
(
t
)
,

a
m
p
;

u
x

(
L
,
t
)
=

υ
x

(
L
,
t
)
=
0

\begin {array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsilon + {u^2}(1 + \alpha - u),} & {{\upsilon _t} = \eta {\upsilon _{xx}} + \sigma u - \gamma \upsilon ,} \\ {u(0,t) = g(t),\quad \upsilon (0,t) = h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \\ \end {array}

\]
where

t
⩾
0

t \geqslant 0

and

0
>
x
>
L
⩽
∞

0 > x > L \leqslant \infty

. Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for

L
L

infinite and the solutions when

L
L

is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.
Codimension 1 orbits and semi-invariants for the representations of an equioriented graph of type 𝐷_{𝑛}
Authors: Abeasis S. 
Published: June 2010

We consider the Dynkin diagram

D
n

{D_n}

equioriented and the variety

Hom
⁡
(

V
1

,

V
3

)
×

Π

1
=
2

n

Hom
⁡
(

V
i

,

V

i
+
1

)

\operatorname {Hom}({V_1},{V_3}) \times \Pi _{1 = 2}^n \operatorname {Hom} ({V_i},{V_{i + 1}})

,

V
j

{V_j}

a vector space over

K
K

, on which the group

G
=

∏

i
=
1

n

GL

(

V
i

)

G = \prod \nolimits _{i = 1}^n {{\text {GL}}} ({V_i})

acts. We determine the maximal orbit and the codim.

1
1

orbits of this action, giving their decomposition in terms of the irreducible representations of

D
n

{D_n}

. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
A general approach to the optimality of minimum distance estimators
Authors: Millar P. W. 
Published: June 2010

Let

Θ
\Theta

be an open subset of a separable Hilbert space, and

ξ
n

(
θ
)

{\xi _n}(\theta )

,

θ
∈
Θ

\theta \in \Theta

, a sequence of stochastic processes with values in a (different) Hilbert space

B
B

. This paper develops an asymptotic expansion and an asymptotic minimax result for "estimates"

θ
^

n

{\hat \theta _n}

defined by

inf
θ

|

ξ
n

(
θ
)

|

=

|

ξ
n

(

θ
^

n

)

|

{\inf _\theta }|{\xi _n}(\theta )| = |{\xi _n}({\hat \theta _n})|

, where

|

⋅

|

| \cdot |

is the norm of

B
B

. The abstract results are applied to study optimality and asymptotic normality of procedures in a number of important practical problems, including simple regression, spectral function estimation, quantile function methods, min-chi-square methods, min-Hellinger methods, minimum distance methods based on

M
M

-functionals, and so forth. The results unify several studies in the literature, but most of the

LAM

{\text {LAM}}

results are new. From the point of view of applications, the entire paper is a sustained essay concerning the problem of fitting data with a reasonable, but relatively simple, model that everyone knows cannot be exact.
Local vanishing properties of solutions of elliptic and parabolic quasilinear equations
Authors: Díaz J. Ildefonso , Véron Laurent 
Published: June 2010

We use a local energy method to study the vanishing property of the weak solutions of the elliptic equation

−
div

A
(
x
,
u
,
D
u
)
+
B
(
x
,
u
,
D
u
)
=
0

- \operatorname {div}\;A(x,u,Du) + B(x,u,Du) = 0

and of the parabolic equation

∂
ψ
(
u
)

/

∂
t
−
div

A

(
t
,
x
,
u
,
D
u
)
+

B

(
t
,
x
,
u
,
D
u
)
=
0

\partial \psi (u)/\partial t - \operatorname {div}\;\mathcal {A}(t,x,u,Du) + \mathcal {B}(t,x,u,Du) = 0

. The results are obtained without any assumption of monotonicity on

A
A

,

B
B

,

A

\mathcal {A}

and

B

\mathcal {B}

.
A rearranged good 𝜆 inequality
Authors: Bagby Richard J. , Kurtz Douglas S. 
Published: June 2010

Let

T
f

Tf

be a maximal Calderón-Zygmund singular integral,

M
f

Mf

the Hardy-Littlewood maximal function, and

w
w

an

A
∞

{A_\infty }

weight. We replace the “good

λ
\lambda

” inequality
\[

w

(

{
x
:

T
f
(
x
)
>
2
λ

and

M
f
(
x
)
≤
ε
λ
}

)

≤
C
(
ε
)
w

(

{
x
:

T
f
(
x
)
>
λ
}

)

w\left ( {\{ x:\,Tf(x) > 2\lambda \,{\text {and}}\,Mf(x) \leq \varepsilon \lambda \} } \right ) \leq C(\varepsilon )w\left ( {\{ x:\,Tf(x) > \lambda \} } \right )

\]
by the rearrangement inequality
\[

(
T
f

)
w
∗

(
t
)
≤
C
(
M
f

)
w
∗

(
t

/

2
)
+
(
T
f

)
w
∗

(
2
t
)

(Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t)

\]
and show that it gives better estimates for

T
f

Tf

. In particular, we obtain best possible weighted

L
p

{L^p}

bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for

(
T
f

)
∗

{(Tf)^ \ast }

.
On sieved orthogonal polynomials. III. Orthogonality on several intervals
Authors: Ismail Mourad E. H. 
Published: June 2010

We introduce two generalizations of Chebyshev polynomials. The continuous spectrum of either is

{
x
:
−
2

c

/

(
1
+
c
)
⩽

T
k

(
x
)
⩽
2

c

/

(
1
+
c
)
}

\{ x: - 2\sqrt c /(1 + c) \leqslant {T_k}(x) \leqslant 2\sqrt c /(1 + c)\}

, where

c
c

is a positive parameter. The weight function of the polynomials of the second kind is

{
1
−
(

(
1
+
c

)
2

/

4
c
)

T
k
2

(
x
)

}

1

/

2

/

|

U

k
−
1

(
x
)

|

{\{ 1 - ({(1 + c)^2}/4\operatorname {c} )T_k^2(x)\} ^{1/2}}/|{U_{k - 1}}(x)|

when

c
⩾
1

c \geqslant 1

. When

c
>
1

c > 1

we pick up discrete masses located at the zeros of

U

k
−
1

(
x
)

{U_{k - 1}}(x)

. The weight function of the polynomials of the first kind is also included. Sieved generalizations of the symmetric Pollaczek polynomials and their

q
q

-analogues are also treated. Their continuous spectra are also the above mentioned set. The

q
q

-analogues include a sieved version of the Rogers

q
q

-ultraspherical polynomials and another set of

q
q

-ultraspherical polynomials discovered by Askey and Ismail. Generating functions and explicit formulas are also derived.
On 𝐾₃ of truncated polynomial rings
Authors: Aisbett Janet 
Published: June 2010

Group homology spectral sequences are used to investigate

K
3

{K_3}

of truncated polynomial rings. If

F
F

is a finite field of odd characteristic, we show that relative

K
2

{K_2}

of the pair

(
F

[
t
]

/

(

t
q

)
,

(

t
k

)
)

(F\left [ t \right ]/({t^q}),\,({t^k}))

, which has been identified by van der Kallen and Stienstra, is isomorphic to

K
3

(
F

[
t
]

/

(

t
k

)
,

(
t
)
)

{K_3}(F\left [ t \right ]/({t^k}),\,(t))

when

q
q

is sufficiently large. We also show that

H
3

(

SL

Z

[
t
]

/

(

t
k

)
;

Z

)
=

Z

k
−
1

⊕

Z

/

24

{H_3}({\text {SL}}\,{\mathbf {Z}}\left [ t \right ]/({t^k});{\mathbf {Z}}) = {{\mathbf {Z}}^{k - 1}} \oplus {\mathbf {Z}}/24

and is isomorphic to the associated

K
3

{K_3}

group modulo an elementary abelian

2
2

-group.
Automorphic images of commutative subspace lattices
Authors: Harrison K. J. , Longstaff W. E. 
Published: June 2010

Let

C
(
H
)

C(H)

denote the lattice of all (closed) subspaces of a complex, separable Hilbert space

H
H

. Let

(

AC)

({\text {AC)}}

be the following condition that a subspace lattice

F

⊆
C
(
H
)

\mathcal {F} \subseteq C(H)

may or may not satisfy: (AC)
\[

F

=
ϕ
(

L

)

for

some

lattice

automorphism

ϕ

of

C
(
H
)

and

some

commutative

subspace

lattice

L

⊆
C
(
H
)
.

\begin {array}{*{20}{c}} {\mathcal {F} = \phi (\mathcal {L})\;{\text {for}}\;{\text {some}}\;{\text {lattice}}\;{\text {automorphism}}\;\phi \;{\text {of}}\;C(H)} \\ {{\text {and}}\;{\text {some}}\;{\text {commutative}}\;{\text {subspace}}\;{\text {lattice}}\;\mathcal {L} \subseteq C(H).} \\ \end {array}

\]
Then

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

A

⊆

B

\mathcal {A} \subseteq \mathcal {B}

for some Boolean algebra subspace lattice

B

⊆
C
(
H
)

\mathcal {B} \subseteq C(H)

with the property that, for every

K
,
L
∈

B

K,L \in \mathcal {B}

, the vector sum

K
+
L

K + L

is closed. If

F

\mathcal {F}

is finite, then

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

F

\mathcal {F}

is distributive and

K
+
L

K + L

is closed for every

K
,
L
∈

F

K,L \in \mathcal {F}

. In finite dimensions

F

\mathcal {F}

satisfies

(

AC

)

({\text {AC}})

if and only if

F

\mathcal {F}

is distributive. Every

F

\mathcal {F}

satisfying

(

AC

)

({\text {AC}})

is reflexive. For such

F

\mathcal {F}

, given vectors

x
,
y
∈
H

x,y \in H

, the solvability of the equation

T
x
=
y

Tx = y

for

T
∈
Alg

F

T \in \operatorname {Alg}\,\mathcal {F}

is investigated.
Stable finitely homogeneous structures
Authors: Cherlin G. , Lachlan A. H. 
Published: June 2010

Let

L
L

be a finite relational language and

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

denote the class of countable

L
L

-structures which are stable and homogeneous. The main result of the paper is that there exists a natural number

c
(
L
)

c(L)

such that for any transitive

M

∈
Hom
⁡
(
L
;
ω
)

\mathcal {M} \in \operatorname {Hom}(L;\omega )

, if

E
E

is a maximal

0
0

-definable equivalence relation on

M

\mathcal {M}

, then either

|

M

/

E

|

>
c
(
L
)

|\mathcal {M}/E| > c(L)

, or

M

/

E

\mathcal {M}/E

is coordinatizable. In an earlier paper the second author analyzed certain subclasses

Hom
⁡
(
L
,
r
)

(
r
>
ω
)

\operatorname {Hom}(L, r)\ (r > \omega )

of

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

for all sufficiently small

r
r

. Thus the earlier analysis now applies to

Hom
⁡
(
L
,
ω
)

\operatorname {Hom}(L,\omega )

.
Isoperimetric inequalities for the least harmonic majorant of |𝑥|^{𝑝}
Authors: Sakai Makoto 
Published: June 2010

Let

D
D

be an open set in the

d
d

-dimensional Euclidean space

R

d

{{\mathbf {R}}^d}

containing the origin

0
0

and let

h

(
p
)

(
x
,
D
)

{h^{(p)}}(x,D)

be the least harmonic majorant of

|

x

|

p

|x{|^p}

in

D
D

, where

0
>
p
>
∞

0 > p > \infty

if

d
⩾
2

d \geqslant 2

and

1
⩽
p
>
∞

1 \leqslant p > \infty

if

d
=
1

d = 1

. We shall be concerned with the following isoperimetric inequalities

h

(
p
)

(
0
,
D

)

1

/

p

⩽
c
r
(
D
)

{h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)

, where

r
(
D
)

r(D)

denotes the volume radius of

D
D

, namely, a ball with radius

r
(
D
)

r(D)

has the same volume as

D
D

has and

c
c

is a constant dependent on

d
d

and

p
p

but independent of

D
D

. We fix

d
d

and denote by

c
(
p
)

c(p)

the infimum of such constants

c
c

. As a function of

p
p

,

c
(
p
)

c(p)

is nondecreasing and satisfies

c
(
p
)
⩾
1

c(p) \geqslant 1

. We shall show (1) there are positive constants

C
1

{C_1}

and

C
2

{C_2}

such that

C
1

p

(
d
−
1
)

/

d

⩽
c
(
p
)
⩽

C
2

p

(
d
−
1
)

/

d

{C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}

for

p
⩾
1

p \geqslant 1

, (2)

c
(
p
)
=
1

c(p) = 1

if

p
⩽
d
+

2

1
−
d

p \leqslant d + {2^{1 - d}}

. Many estimations of

h

(
p
)

(
0
,
D
)

{h^{(p)}}(0,D)

and their applications are also given.
A dimension formula for Hermitian modular cusp forms of degree two
Authors: Eie Min King 
Published: June 2010

An explicit dimension formula for the vector space of Hermitian modular cusp forms of degree two with respect to the modular group

Γ
2

(

Z

[
i
]
)
=
SU
⁡
(
2
,
2
)
∩

M
4

(

Z

[
i
]
)

{\Gamma _2}({\mathbf {Z}}[i]) = \operatorname {SU} (2,2) \cap {M_4}({\mathbf {Z}}[i])

is obtained via the Selberg trace formula and its arithmetic properties. Also, a generating function for the graded ring of Hermitian cusp forms of degree two is given.
Where does the 𝐿^{𝑝}-norm of a weighted polynomial live?
Authors: Mhaskar H. N. , Saff E. B. 
Published: June 2010

For a general class of nonnegative weight functions

w
(
x
)

w(x)

having bounded or unbounded support

Σ
⊂

R

\Sigma \subset {\mathbf {R}}

, the authors have previously characterized the smallest compact set

S

w

{\mathfrak {S}_w}

, having the property that for every

n
=
1
,

2
,
…

n = 1,\,2, \ldots

and every polynomial

P
P

of degree

⩽
n

\leqslant n

,
\[

|

|

[
w
(
x
)

]
n

P
(
x
)

|

|

L
∞

(
Σ
)

=

|

|

[
w
(
x
)

]
n

P
(
x
)

|

|

L
∞

(

S

w

)

||{[w(x)]^n}P(x)|{|_{{L^\infty }(\Sigma )}} = ||{[w(x)]^n}P(x)|{|_{{L^\infty }({\mathfrak {S}_w})}}

\]
. In the present paper we prove that, under mild conditions on

w
w

, the

L
p

{L^p}

-norms

(
0
>
p
>
∞
)

(0 > p > \infty )

of such weighted polynomials also "live" on

S

w

{\mathfrak {S}_w}

in the sense that for each

η
>
0

\eta > 0

there exist a compact set

Δ
\Delta

with Lebesgue measure

m
(
Δ
)
>
η

m(\Delta ) > \eta

and positive constants

c
1

{c_1}

,

c
2

{c_2}

such that
\[

|

|

w
n

P

|

|

L
p

(
Σ
)

⩽
(
1
+

c
1

exp
⁡
(
−

c
2

n
)
)

|

|

w
n

P

|

|

L
p

(

S

w

∪
Δ
)

||{w^n}P|{|_{{L^p}(\Sigma )}} \leqslant (1 + {c_1}\exp ( - {c_2}n))||{w^n}P|{|_{{L^p}({\mathfrak {S}_w} \cup \Delta )}}

\]
. As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.
Mixed norm estimates for certain means
Authors: Börjeson Lennart 
Published: June 2010

We obtain estimates of the mean
\[

F
x
γ

(
t
)
=

C
γ

∫

|

y

|

>
1

(
1
−

|

y

|

2

)

γ

f
(
x
−
t
y
)

d
y

F_x^\gamma (t) = {C_\gamma }\int _{|y| > 1} {{{(1 - |y{|^2})}^\gamma }f(x - ty)\,dy}

\]
in mixed Lebesgue and Sobolev spaces. They generalize earlier estimates of the spherical mean

F
x

−
1

(
t
)
=
C

∫

S

n
−
1

f
(
x
−
t
y
)

d
S
(
y
)

F_x^{ - 1}(t) = C\;\int _{{S^{n - 1}}} {f(x - ty)\,dS(y)}

and of solutions of the wave equation

Δ
x

u
=

∂
2

u

/

∂

t
2

{\Delta _x}u = {\partial ^2}u/\partial {t^2}

.
Sieving the positive integers by small primes
Authors: Goldston D. A. , McCurley Kevin S. 
Published: June 2010

Let

Q
Q

be a set of primes that has relative density

δ
\delta

among the primes, and let

ϕ
(
x
,

y
,

Q
)

\phi (x,\,y,\,Q)

be the number of positive integers

⩽
x

\leqslant x

that have no prime factor

⩽
y

\leqslant y

from the set

Q
Q

. Standard sieve methods do not seem to give an asymptotic formula for

ϕ
(
x
,

y
,

Q
)

\phi (x,\,y,\,Q)

in the case that

1
2

⩽
δ
>
1

\tfrac {1}{2} \leqslant \delta > 1

. We use a method of Hildebrand to prove that
\[

ϕ
(
x
,
y
,
Q
)
∼
x
f
(
u
)

∏

p
>
y

p
∈
Q

(

1
−

1
p

)

\phi (x,y,Q) \sim x f(u) \prod _{\substack {p > y\\p \in Q}} {\left ( {1 - \frac {1}{p}} \right )}

\]
as

x
→
∞

x \to \infty

, where

u
=

log
⁡
x

log
⁡
y

u = \frac {{\log x}}{{\log y}}

and

f
(
u
)

f(u)

is defined by
\[

u
δ

f
(
u
)
=

{

e

γ
δ

Γ
(
1
−
δ
)

,

a
m
p
;

0
>
u
⩽
1
,

e

γ
δ

Γ
(
1
−
δ
)

+
δ

∫
0

u
−
1

f
(
t
)

(
1
+
t
)

δ
−
1

d
t
,

a
m
p
;

u
>
1.

{u^\delta }f(u) = \left \{ {\begin {array}{*{20}{c}} {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}},} \hfill & {0 > u \leqslant 1,} \hfill \\ {\frac {{{e^{{\gamma ^\delta }}}}}{{\Gamma (1 - \delta )}} + \delta \int _0^{u - 1} {f(t){{(1 + t)}^{\delta - 1}}\;dt,} } \hfill & {u > 1.} \hfill \\ \end {array} } \right .

\]
This may also be viewed as a generalization of work by Buchstab and de Bruijn, who considered the case where

Q
Q

consisted of all primes.
Linear supergroup actions. I. On the defining properties
Authors: Sánchez-Valenzuela Oscar Adolfo 
Published: June 2010

This paper studies the notions of linearity and bilinearity in the category of supermanifolds. Following the work begun by [OASV2], we deal with supermanifoldifications of supervector spaces. The

R

1

|

1

{{\mathbf {R}}^{1|1}}

-module operations are defined componentwise. The linearity and bilinearity properties are stated by requiring commutativity of some appropriate diagrams of supermanifold morphisms. It is proved that both linear and bilinear supermanifold morphisms are completely determined by their underlying continuous maps, which in turn have to be linear (resp., bilinear) in the usual sense. It is observed that whereas linear supermanifold morphisms are vector bundle maps, bilinear supermanifold morphisms are not. A natural generalization of the bilinear evaluation map

Hom
⁡
(
V
,

W
)
×
V
→
W

(
(
F
,

v
)
↦
F
(
v
)
)

\operatorname {Hom} (V,\,W) \times V \to W\;((F,\,v) \mapsto F(v))

is given and some applications pointing toward the notions of linear supergroup actions and adjoint and coadjoint actions are briefly discussed.
A very singular solution of a quasilinear degenerate diffusion equation with absorption
Authors: Peletier L. A. , Wang Jun Yu 
Published: June 2010

The object of this paper is to study the existence of a nonnegative solution of the Cauchy problem
\[

u
t

=
div
⁡
(

|

∇
u

|

p
−
2

∇
u
)
−

u
q

,

u
(
x
,

0
)
=
0

if

x
≠
0
,

{u_t} = \operatorname {div} (|\nabla u{|^{p - 2}}\nabla u) - {u^q},\qquad u(x,\,0) = 0\quad {\text {if}}\;x \ne 0,

\]
which is more singular at

(
0
,

0
)

(0,\,0)

than the fundamental solution of the corresponding equation without the absorption term.
On zeros of a system of polynomials and application to sojourn time distributions of birth-and-death processes
Authors: Sato Ken-iti 
Published: June 2010

Zeros of the following system of polynomials are considered:
\[

{

P
0

(
x
)
=
1
,

P
1

(
x
)
=

B
0

−

A
0

x
,

P

n
+
1

(
x
)
=
(

B
n

−

A
n

x
)

P
n

(
x
)
−

C
n

P

n
−
1

(
x
)

for

n
⩾
1.

\left \{ \begin {gathered} {P_0}(x) = 1, \hfill \\ {P_1}(x) = {B_0} - {A_0}x, \hfill \\ {P_{n + 1}}(x) = ({B_n} - {A_n}x){P_n}(x) - {C_n}{P_{n - 1}}(x)\quad {\text {for}}\;n \geqslant 1. \hfill \\ \end {gathered} \right .

\]
Numbers of positive and negative zeros are determined and a separation property of the zeros of

P
m

(
x
)

{P_m}(x)

and

P
n

(
x
)

{P_n}(x)

is proved under the condition that

C
n

>
0

{C_n} > 0

and

P
n

(
0
)
>
0

{P_n}(0) > 0

for every

n
n

. No condition is imposed on

A
n

{A_n}

. These results are applied to determination of the distribution of a sojourn time with general (not necessarily positive) weight function for a birth-and-death process up to a first passage time. Unimodality and infinite divisibility of the distribution follow.