bluuebunny/crossref_metadata_2025_split huggingface dataset:
A two weight weak type inequality for fractional integrals
Authors: Sawyer Eric 
Published: June 2010

For

1
>
p
⩽
q
>
∞
,
0
>
α
>
n

1 > p \leqslant q > \infty ,0 > \alpha > n

and

w
(
x
)
,
υ
(
x
)

w(x),\upsilon (x)

nonnegative weight functions on

R
n

{R^n}

we show that the weak type inequality
\[

∫

{

T
α

f
>
λ
}

w
(
x
)

d
x
⩽
A

λ

−
q

(
∫

|

f
(
x
)

|

p

υ
(
x
)

d
x
)

q

/

p

\int _{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}}

\]
holds for all

f
⩾
0

f \geqslant 0

if and only if
\[

∫
Q

[

T
α

(

χ
Q

w
)

(
x
)

]

p
′

υ
(
x

)

1
−

p
′

d
x
⩽
B

(

∫
Q

w
)

p
′

/

q
′

>
∞

\int _Q\,[{T_\alpha }({\chi _Q}w)\,(x)]^{p’}\upsilon (x)^{1 - p’}\,dx \leqslant B\left ( \int _Qw \right )^{p’/q’} > \infty

\]
for all cubes

Q
Q

in

R
n

{R^n}

. Here

T
α

{T_\alpha }

denotes the fractional integral of order

α
,

T
α

f
(
x
)
=
∫

|

x
−
y

|

α
−
n

f
(
y
)

d
y

\alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y)\,dy

. More generally we can replace

T
α

{T_\alpha }

by any suitable convolution operator with radial kernel decreasing in

|

x

|

|x|

.
On the paths of symmetric stable processes
Authors: Davis Burgess 
Published: June 2010

It is shown that if

X
(
t
)
,
t
⩾
0

X(t), t \geqslant 0

, is a symmetric stable process of index

α
,
0
>
α
>
2

\alpha , 0 > \alpha > 2

, then

sup
t

lim

inf

h
↓
0

(
X
(
t
+
h
)
−
X
(
t
)
)

h

−
1

/

α

=
∞

\sup _t \lim \inf _{h \downarrow 0} (X(t + h) - X(t))h^{-1/\alpha } = \infty

a.s. This settles a question of Fristedt about strictly stable subordinators.
Stable viscosities and shock profiles for systems of conservation laws
Authors: Pego Robert L. 
Published: June 2010

Wide classes of high order "viscosity" terms are determined, for which small amplitude shock wave solutions of a nonlinear hyperbolic system of conservation laws

u
t

+
f

(
u

)
x

=
0

{u_t} + f{(u)_x} = 0

are realized as limits of traveling wave solutions of a dissipative system

u
t

+
f

(
u

)
x

=
ν

(

D
1

u
x

)
x

+
⋯
+

ν
n

(

D
n

u

(
n
)

)
x

{u_t} + f{(u)_x} = \nu {({D_1}{u_x})_x} + \cdots + {\nu ^n}{({D_n}{u^{(n)}})_x}

. The set of such "admissible" viscosities includes those for which the dissipative system satisfies a linearized stability condition previously investigated in the case

n
=
1

n = 1

by A. Majda and the author. When

n
=
1

n = 1

we also establish admissibility criteria for singular viscosity matrices

D
1

(
u
)

{D_1}(u)

, and apply our results to the compressible Navier-Stokes equations with viscosity and heat conduction, determining minimal conditions on the equation of state which ensure the existence of the "shock layer" for weak shocks.
The heat equation with a singular potential
Authors: Baras Pierre , Goldstein Jerome A. 
Published: June 2010

Of concern is the singular problem

∂
u

/

∂
t
=
Δ
u
+
(
c

/

|

x

|

2

)

u
+
f
(
t
,
x
)
,
u
(
x
,
0
)
=

u

0

(
x
)

\partial u/\partial t = \Delta u + (c/|x{|^2})\,u + f(t,x), u(x,0) = u_{0}(x)

, and its generalizations. Here

c
⩾
0
,
x
∈

R

N

,
t
>
0

c \geqslant 0,x \in {{\mathbf {R}}^N},t > 0

, and

f
f

and

u
0

{u_0}

are nonnegative and not both identically zero. There is a dimension dependent constant

C

∗

(
N
)

{C_{\ast } }(N)

such that the problem has no solution for

c
>

C

∗

(
N
)

c > {C_{\ast } }(N)

. For

c
⩽

C

∗

(
N
)

c \leqslant {C_{\ast } }(N)

necessary and sufficient conditions are found for

f
f

and

u
0

{u_0}

so that a nonnegative solution exists.
A method of lines for a nonlinear abstract functional evolution equation
Authors: Kartsatos A. G. , Parrott M. E. 
Published: June 2010

Let

X
X

be a real Banach space with

X
∗

{X^\ast }

uniformly convex. A method of lines is introduced and developed for the abstract functional problem (E)
\[

u
′
(
t
)
+
A
(
t
)
u
(
t
)
=
G
(
t
,

u
t

)
,

u
0

=
ϕ
,

t
∈
[
0
,
T
]
.

u\prime (t) + A(t)u(t) = G(t,{u_t}), \quad {u_0} = \phi , \quad t \in [0,T].

\]
The operators

A
(
t
)
:
D
⊂
X
→
X

A(t):D \subset X \to X

are

m
m

-accretive and

G
(
t
,
ϕ
)

G(t,\phi )

is a global Lipschitzian-like function in its two variables. Further conditions are given for the convergence of the method to a strong solution of (E). Recent results for perturbed abstract ordinary equations are substantially improved. The method applies also to large classes of functional parabolic problems as well as problems of integral perturbations. The method is straightforward because it avoids the introduction of the operators

A
^

(
t
)

\hat A(t)

and the corresponding use of nonlinear evolution operator theory.
𝐵𝑀𝑂(𝜌) and Carleson measures
Authors: Smith Wayne Stewart 
Published: June 2010

This paper concerns certain generalizations of

BMO

{\text {BMO}}

, the space of functions of bounded mean oscillation. Let

ρ
\rho

be a positive nondecreasing function on

(
0
,
∞
)

(0,\infty )

with

ρ
(
0
+
)
=
0

\rho (0 + ) = 0

. A locally integrable function on

R

m

{{\mathbf {R}}^m}

is said to belong to

BMO

(
ρ
)

{\text {BMO}}(\rho )

if its mean oscillation over any cube

Q
Q

is

O
(
ρ
(
l
(
Q
)
)
)

O(\rho (l(Q)))

, where

l
(
Q
)

l(Q)

is the edge length of

Q
Q

. Carleson measures are known to be closely related to

BMO

{\text {BMO}}

. Generalizations of these measures are shown to be similarly related to the spaces

BMO

(
ρ
)

{\text {BMO}}(\rho )

. For a cube

Q
Q

in

R

m

,

|

Q

|

{{\mathbf {R}}^m},\;|Q|

denotes its volume and

R
(
Q
)

R(Q)

is the set

{
(
x
,
y
)
∈

R

+

m
+
1

:
x
∈
Q
,

0
>
y
>
l
(
Q
)
}

\{ (x,y) \in {\mathbf {R}}_ + ^{m + 1}:x \in Q,\;0 > y > l(Q)\}

. A measure

μ
\mu

on

R

+

m
+
1

{\mathbf {R}}_ + ^{m + 1}

is called a

ρ
\rho

-Carleson measure if

|

μ

|

(
R
(
Q
)
)
=
O
(
ρ
(
l
(
Q
)
)

|

Q

|

)

|\mu |(R(Q)) = O(\rho (l(Q))|Q|)

, for all cubes

Q
Q

. L. Carleson proved that a compactly supported function in

BMO

{\text {BMO}}

can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving

BMO

(
ρ
)

{\text {BMO}}(\rho )

and

ρ
\rho

-Carleson measures is proved for a broad class of growth functions, and this is used to represent

BMO

(
ρ
)

{\text {BMO}}(\rho )

as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson’s theorem. Another characterization of

BMO

(
ρ
)

{\text {BMO}}(\rho )

using

ρ
\rho

-Carleson measures is a corollary. This result generalizes a characterization of

BMO

{\text {BMO}}

due to C. Fefferman. Finally, an atomic decomposition of the predual of

BMO

(
ρ
)

{\text {BMO}}(\rho )

is given.
On coupled multiparameter nonlinear elliptic systems
Authors: Cantrell Robert Stephen 
Published: June 2010

This paper considers the system of nonlinear Dirichlet boundary value problems
\[

{

L
u
(
x
)
=
λ
f
(
u
(
x
)
,
v
(
x
)
)

L
v
(
x
)
=
μ
g
(
u
(
x
)
,
v
(
x
)
)

}

,

x
∈
Ω
,

\left \{ \begin {gathered} Lu(x) = \lambda f(u(x),v(x)) \hfill \\ Lv(x) = \mu g(u(x),v(x)) \hfill \\ \end {gathered} \right \},\qquad x \in \Omega ,

\]
a bounded domain in

R

n

{{\mathbf {R}}^n}

. Here

L
L

is a strongly, uniformly elliptic linear partial differential operator,

λ
\lambda

,

μ
\mu

are real parameters, and

f
f

,

g
:

R

2

→
R

g:{{\mathbf {R}}^2} \to R

are smooth with
\[

f
(
0
,
0
)
=
0
=
g
(
0
,
0
)
.

f(0,0) = 0 = g(0,0).

\]
A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.
Concavity of solutions of the porous medium equation
Authors: Bénilan Philippe , Vázquez Juan Luis 
Published: June 2010

We consider the problem
\[

(

P

)

{

u
t

=

(

u
m

)

x
x

,

u
(
x
,
0
)
=

u
0

(
x
)

with

x
∈

R

,

t
>
0

for

x
∈

R

\left ( {\text {P}} \right )\quad \quad \left \{ {\begin {array}{*{20}{c}} {{u_t} = {{({u^m})}_{xx}},} \\ {u(x,0) = {u_0}(x)} \\ \end {array} } \right .\quad \begin {array}{*{20}{c}} {{\text {with}}\,x \in {\mathbf {R}},\,t > 0} \\ {{\text {for}}\,x \in {\mathbf {R}}} \\ \end {array}

\]
where

m
>
1

m > 1

and

u
0

{u_0}

is a continuous, nonnegative function that vanishes outside an interval

(
a
,

b
)

(a,\,b)

and such that

(

u
0

m
−
1

)
≤
−
C
≤
0

(u_0^{m - 1}) \leq - C \leq 0

in

(
a
,

b
)

(a,\,b)

. Using a Trotter-Kato formula we show that the solution conserves the concavity in time: for every

t
>
0
,

u
(
x
,
t
)

t > 0,\,u(x,t)

vanishes outside an interval

Ω
(
t
)
=
(

ζ
1

(
t
)
,

ζ
2

(
t
)
)

\Omega (t) = ({}_{\zeta 1}(t),\,{}_{\zeta 2}(t))

and
\[

(

u

m
−
1

)

x
x

≤
−

C

1
+
C
(
m
(
m
+
1
)

/

(
m
−
1
)
)
t

{({u^{m - 1}})_{xx}} \leq - \frac {C} {{1 + C(m(m + 1)/(m - 1))t}}

\]
in

Ω
(
t
)

\Omega (t)

. Consequently the interfaces

x

=

ζ
i

(
t
)

x{ = _{\zeta i}}(t)

,

i
=
1
,

2

i = 1,\,2

, are concave curves. These results also give precise information about the large time behavior of solutions and interfaces.
Quadratic geometry of numbers
Authors: Schlickewei Hans Peter , Schmidt Wolfgang M. 
Published: June 2010

We give upper bounds for zeros of quadratic forms. For example we prove that for any nondegenerate quadratic form

F

(

x
1

,
…
,

x
n

)

\mathfrak {F}({x_1}, \ldots ,\,{x_n})

with rational integer coefficients which vanishes on a

d
d

-dimensional rational subspace

(
d
>
0
)

(d > 0)

there exist sublattices

Γ
0

,

Γ
1

,

…

,

Γ

n
−
d

{\Gamma _0},\,{\Gamma _1},\, \ldots \,,{\Gamma _{n - d}}

of

Z

n

{\mathbf {Z}^n}

of rank

d
d

, on which

F

\mathfrak {F}

vanishes, with the following properties:
\[

rank

(

Γ
0

∩

Γ
i

)
=
d
−
1
,

rank

(

Γ
0

∪

Γ
1

∪
⋯
∪

Γ

n
−
d

)
=
n

{\text {rank}}({\Gamma _0} \cap {\Gamma _i}) = d - 1,\quad {\text {rank}}({\Gamma _0} \cup {\Gamma _1} \cup \cdots \cup {\Gamma _{n - d}}) = n

\]
and
\[

(
det

Γ
0

)

n
−
d

det

Γ
1

⋯
det

Γ

n
−
d

≪

F

(
n
−
d
)

2

{(\det \,{\Gamma _0})^{n - d}}\det \,{\Gamma _1} \cdots \det \,{\Gamma _{n - d}} \ll {F^{{{(n - d)}^2}}}

\]
, where

F
F

is the maximum modulus of the coefficients of

F

\mathfrak {F}

.
𝐾-theory and right ideal class groups for HNP rings
Authors: Hodges Timothy J. 
Published: June 2010

Let

R
R

be an hereditary Noetherian prime ring, let

S
S

be a "Dedekind closure" of

R
R

and let

T

\mathcal {T}

be the category of finitely generated

S
S

-torsion

R
R

-modules. It is shown that for all

i
≥
0

i \geq 0

, there is an exact sequence

0
→

K
i

(

T

)
→

K
i

(
R
)
→

K
i

(
S
)
→
0

0 \to {K_i}(\mathcal {T}) \to {K_i}(R) \to {K_i}(S) \to 0

. If

i
=
0

i = 0

, or

R
R

has finitely many idempotent ideals then this sequence splits. A notion of "right ideal class group" is then introduced for hereditary Noetherian prime rings which generalizes the standard definition of class group for hereditary orders over Dedekind domains. It is shown that there is a decomposition

K
0

(
R
)
≅

Cl

(
R
)
⊕
F

{K_0}(R) \cong {\text {Cl}}(R) \oplus F

where

F
F

is a free abelian group whose rank depends on the number of idempotent maximal ideals of

R
R

. Moreover there is a natural isomorphism

Cl

(
R
)
≅

Cl

(
S
)

{\text {Cl}}(R) \cong {\text {Cl}}(S)

and this decomposition corresponds closely to the splitting of the above exact sequence for

K
0

{K_0}

.
Homological stability for 𝑂_{𝑛,𝑛} over a local ring
Authors: Betley Stanisław 
Published: June 2010

Let

R
R

be a local ring,

V

2
n

{V^{2n}}

a free module over

R
R

of rank

2
n

2n

and

q
q

a bilinear form on

V

2
n

{V^{2n}}

which has in some basis the matrix

|

0

a
m
p
;
1

1

a
m
p
;
0

|

\left | {\begin {array}{*{20}{c}} 0 & 1 \\ 1 & 0 \\ \end {array} } \right |\,

. Let

O

n
,
n

{O_{n,n}}

be the group of automorphisms of

V

2
n

{V^{2n}}

which preserve

q
q

. We prove the following theorem: if

n
n

is big enough with respect to

k
k

then the inclusion homomorphism

i
:

O

n
,
n

→

O

n
+
1
,
n
+
1

i:{O_{n,n}} \to {O_{n + 1,n + 1}}

induces an isomorphism

i

∗

:

H
k

(

O

n
,
n

;

Z
)
→

H
k

(

O

n
+
1
,
n
+
1

;
Z
)

{i_{\ast }}:{H_k}({O_{n,n}};\,Z) \to {H_k}({O_{n + 1,n + 1}};Z)

.
Existence domains of holomorphic functions of restricted growth
Authors: Jarnicki M. , Pflug P. 
Published: June 2010

The paper presents a large class of domains

G
G

of holomorphy in

C

n

{{\mathbf {C}}^n}

such that, for any

N
>
0

N > 0

, there exists a nonextendable holomorphic function

f
f

on

G
G

with

|

f

|

δ
G
N

|f|\delta _G^N

bounded where

δ
G

(
z
)
:=
min
(

(
1
+

|

z

|

2

)

−
1

/

2

,

dist
⁡
(
z
,

∂
G
)
)

{\delta _G}(z): = \min ({(1 + |z{|^2})^{ - 1 / 2}},\,\operatorname {dist} (z,\,\partial G))

. Any fat Reinhardt domain of holomorphy belongs to this class. On the other hand we characterize those Reinhardt domains of holomorphy which are existence domains of bounded holomorphic functions.
Extremal analytic discs with prescribed boundary data
Authors: Chang Chin-Huei , Hu M. C. , Lee Hsuan-Pei 
Published: June 2010

This paper concerns the existence and uniqueness of extremal analytic discs with prescribed boundary data in a bounded strictly linearly convex domain

D
D

in

C

n

{{\mathbf {C}}^n}

. We prove that for any two distinct points

p
p

,

q
q

in

∂
D

\partial D

(respectively,

p
∈
∂
D

p \in \partial D

and a vector

v
v

such that

−
1

v
∈

T
p

(
∂
D
)

\sqrt { - 1} v \in {T_p}(\partial D)

and

⟨
v
,

ν
¯

(
p
)
⟩
=

∑
1
n

v
j

ν
¯

j

(
p
)
>
0

\langle v,\,\overline \nu (p)\rangle = \sum \nolimits _1^n {{v_j}{{\overline \nu }_j}(p) > 0}

where

ν
(
p
)

\nu (p)

is the outward normal to

∂
D

\partial D

at

p
p

) there exists an extremal analytic disc

f
f

passing through

p
p

,

q
q

if

∂
D
∈

C
k

\partial D \in {C^k}

,

k
⩾
3

k \geqslant 3

(respectively,

f
(
1
)
=
p

f(1) = p

,

f
′

(
1
)
=
v

f’ (1) = v

if

∂
D
∈

C
k

\partial D \in {C^k}

,

k
⩾
14

k \geqslant 14

). Consequently, we can foliate

D
¯

\overline D

with these extremal analytic discs.
Subordination families and extreme points
Authors: Abu-Muhanna Yusuf , Hallenbeck D. J. 
Published: June 2010

Let

s
(
F
)

s(F)

denote the set of functions subordinate to a univalent function

F
F

in

Δ
\Delta

the unit disk. Let

B
0

{B_0}

denote the set of functions

ϕ
(
z
)

\phi (z)

analytic in

Δ
\Delta

satisfying

|

ϕ
(
z
)

|

>
1

|\phi (z)| > 1

and

ϕ
(
0
)
=
0

\phi (0) = 0

. We prove that if

f
=
F
∘
ϕ

f = F \circ \phi

is an extreme point of

s
(
F
)

s(F)

, then

ϕ
\phi

is an extreme point of

B
0

{B_0}

. Let

D
=
F
(
s
)

D = F(s)

and

λ
(
w
,

∂
D
)

\lambda (w,\,\partial D)

denote the distance between

w
w

and

∂
D

\partial D

(boundary of

D
D

). We also prove that if

ϕ
\phi

is an extreme point of

B
0

{B_0}

and

|

ϕ
(

e

i
t

)

|

>
1

|\phi ({e^{it}})| > 1

for almost all

t
t

, then

∫
0

2
π

log
⁡
λ
(
F
(
ϕ
(

e

i
t

)

e

i
θ

)
,

∂
D
)

d
t
=
−
∞

\int _0^{2\pi } {\log \lambda (F(\phi ({e^{it}}){e^{i\theta }}),\,\partial D)\,dt = - \infty }

for almost all

θ
\theta

.
The blow-up surface for nonlinear wave equations with small spatial velocity
Authors: Friedman Avner , Oswald Luc 
Published: June 2010

Consider the Cauchy problem for

u

t
t

−

ε
2

Δ
u
=
f
(
u
)

{u_{tt}} - {\varepsilon ^2}\Delta u = f(u)

in space dimension

⩽
3

\leqslant 3

where

f
(
u
)

f(u)

is superlinear and nonnegative. The solution blows up on a surface

t
=

ϕ
ε

(
x
)

t = {\phi _\varepsilon }(x)

. Denote by

t
=
ϕ
(
x
)

t = \phi (x)

the blow-up surface corresponding to

v
=
f
(
v
)

v = f(v)

. It is proved that

|

ϕ
ε

(
x
)
−
ϕ
(
x
)

|

⩽
C

ε
2

|{\phi _\varepsilon }(x) - \phi (x)| \leqslant C{\varepsilon ^2}

,

|

∇
(

ϕ
ε

(
x
)
−
ϕ
(
x
)
)

|

⩽
C

ε
2

|\nabla ({\phi _\varepsilon }(x) - \phi (x))| \leqslant C{\varepsilon ^2}

in a neighborhood of any point

x
0

{x_0}

where

ϕ
(

x
0

)
>
∞

\phi ({x_0}) > \infty

.
Additive cohomology operations
Authors: Duflot Jeanne 
Published: June 2010

The bigraded group

{

H
i

(

Σ
n

,

Z

/

p
)
}

\{ {H_i}({\Sigma _n},{\mathbf {Z}}/p)\}

becomes a Hopf algebra, if multiplication is induced by restriction, and comultiplication is induced by transfer. Using Steenrod’s method of considering elements of this bigraded group as

mod

-

p

\bmod {\text { - }}p

cohomology operations, the primitives of this Hopf algebra correspond to additive cohomology operations. In this paper we use the results known about the homology and cohomology of the symmetric groups and the operations they induce in

mod

-

p

\bmod {\text { - }}p

cohomology to write down two (additive) bases of the bigraded vector space of primitives of the above Hopf algebra.
Numerical analysis of boundary-layer problems in ordinary differential equations
Authors: Murphy W. D. 
Published: June 2010

We categorize some of the finite-difference methods that can be used to treat the initial-value problem for the boundary-layer differential equation
\[

(
1
)

μ

y
′

=
f
(
y
,
x
)
;
y
(
0
)
=

y
0

.

(1){\text { }}\mu y’ = f(y,x);y(0) = {y^0}.

\]
These methods take the form
\[

(
2
)

∑

i
=
0

k

α
i

Y

n
+
i

=

h

1
−
γ

∑

i
=
0

k

β
i

f

(

Y

n
+
i

,

x

n
+
i

)
+

R
n

,

(2){\text { }}\sum \limits _{i = 0}^k {{\alpha _i}{Y_{n + i}} = {h^{1 - \gamma }}} \sum \limits _{i = 0}^k {{\beta _i}f} ({Y_{n + i}},{x_{n + i}}) + {R_n},

\]
where

α
ν

{\alpha _\nu }

and

β
ν

(
ν
=
0
,
1
,
⋯
,
k
)

{\beta _\nu }(\nu = 0,1, \cdots ,k)

denote real constants which do not depend upon

n
,

R
n

n,{R_n}

is the round-off error,

μ
=

h
r

,
0
>
γ
>
1

\mu = {h^r},0 > \gamma > 1

, and

h
h

is the mesh size. We define a new kind of stability called

μ
\mu

stability and prove that under certain conditions

μ
\mu

stability implies convergence of the difference method. We investigate

μ
\mu

stability and the optimal methods which it allows, i.e., methods of maximum accuracy. The idea of relating

μ
\mu

to

h
h

allows us to study the nature of the difference equation for very small

μ
\mu

. We can, however, look at this in another way. Given a differential equation in the form of Eq. (1) we ask how can we choose

h
h

so that the associated difference equation will give an accurate approximation. If

μ
\mu

is sufficiently small, choose

h
h

by the formula

h
=

μ

1

/

γ

h = {\mu ^{1/\gamma }}

where

0
>
γ
>
1

0 > \gamma > 1

.
The Riemann hypothesis and pseudorandom features of the Möbius sequence
Authors: Good I. J. , Churchhouse R. F. 
Published: June 2010

A study of the cumulative sums of the Möbius function on the Atlas Computer of the Science Research Council has revealed certain statistical properties which lead the authors to make a number of conjectures. One of these is that any conjecture of the Mertens type, viz.
\[

|

M
(
N
)

|

=

|

∑

n
=
1

N

μ
(
n
)

|

>
k

(

√
N

)

|M(N)| = \left | {\sum \limits _{n = 1}^N {\mu (n)} } \right | > k\left ( {\surd N} \right )

\]
where

k
k

is any positive constant, is false, and indeed the authors conjecture that
\[

Lim

sup

{

M
(
x
)

(
x
log
⁡
log
⁡
x
)

−
1

/

2

}

=

√

(

12

)

/

√

(

12

)

π

π

{\text {Lim}}\sup \left \{ {M(x){{(x\log \log x)}^{ - 1/2}}} \right \} = {{\surd \left ( {12} \right )} \left / {\vphantom {{\surd \left ( {12} \right )} \pi }} \right . \pi }

\]
.
Osculatory interpolation
Authors: Kahng S. W. 
Published: June 2010

An explicit method of osculatory interpolation with a function of the form

R
(
x
)
=

a
m
p
;

f

00

(

a
0

0

+

g
0

(
x
)

f
0

1

(

a
0

1

+

g
0

(
x
)

f
0

2

(

a
0

2

+
⋯
+

g
0

(
x
)

a
m
p
;
⋅

f

0
,

m
0

(

a

0
,

m
0

+

g
0

(
x
)

f

10

(

a

10

+

g
1

(
x
)

f

11

(

a

11

a
m
p
;
+
⋯
+

g
1

(
x
)

f

1
,

m
1

(

a

1
,

m
1

+

g
1

(
x
)

f

20

(

a

20

+
⋯
+

g

n
−
1

(
x
)

a
m
p
;
⋅

f

n
0

(

a

n
0

+

g
n

(
x
)

f

n
1

(

a

n
1

+
⋯
+

g
n

(
x
)

f

n
,

m
n

(

a

n
,

m
n

)
)
⋯
)

\begin{align*} R(x) = & {f_{00}}({a_0}_0 + {g_0}(x){f_0}_1({a_0}_1 + {g_0}(x){f_0}_2({a_0}_2 + \cdots + {g_0}(x) \\ & \cdot f_{0, m_0}(a_{0, m_0} + g_0(x)f_{10}(a_{10} + g_1(x)f_{11}(a_{11} \\ & + \cdots + g_1(x)f_{1,m_1}(a_{1,m_1} + g_1(x)f_{20}(a_{20} + \cdots + {g_{n - 1}}(x) \\ & \cdot f_{n0}(a_{n0} + g_n(x)f_{n1}(a_{n1} + \cdots + g_n(x)f_{n, m_n}(a_{n,m_n})) \cdots ) \end{align*}

is described. Error terms for the interpolation are determined.
A finite element collocation method for quasilinear parabolic equations
Authors: Douglas Jim , Dupont Todd 
Published: June 2010

Let the parabolic problem

c
(
x
,
t
,
u
)

u
t

=
a
(
x
,
t
,
u
)

u

x
x

+
b
(
x
,
t
,
u
,

u
x

)
,
0
>
x
>
1
,
0
>
t
≦
T
,
u
(
x
,
0
)
=
f
(
x
)
,
u
(
0
,
t
)
=

g
0

(
t
)
,
u
(
1
,
t
)
=

g
1

(
t
)

c(x,t,u){u_t} = a(x,t,u){u_{xx}} + b(x,t,u,{u_x}),0 > x > 1,0 > t \leqq T,u(x,0) = f(x),u(0,t) = {g_0}(t),u(1,t) = {g_1}(t)

, be solved approximately by the continuous-time collocation process based on having the differential equation satisfied at Gaussian points

ξ

i
,
1

{\xi _{i,1}}

and

ξ

i
,
2

{\xi _{i,2}}

in subintervals

(

x

i
−
1

,

x
i

)

({x_{i - 1}},{x_i})

for a function

U
:
[
0
,
T
]
→

H

3

U:[0,T] \to {\mathcal {H}_3}

, the class of Hermite piecewise-cubic polynomial functions with knots

0
=

x
0

>

x
1

>
⋯
>

x
n

=
1

0 = {x_0} > {x_1} > \cdots > {x_n} = 1

. It is shown that

u
−
U
=
O
(

h
4

)

u - U = O({h^4})

uniformly in x and t, where

h
=
max
(

x
i

−

x

i
−
1

)

h = \max ({x_i} - {x_{i - 1}})

.
Existence of Gauss interpolation formulas for the one-dimensional heat equation
Authors: Barrow David L. 
Published: June 2010

Let

C
=
{
(
x
(
s
)
,
t
(
s
)
)
:
a
⩽
s
⩽
b
}

C = \{ (x(s),t(s)):a \leqslant s \leqslant b\}

be a Jordan arc in the x-t plane satisfying

(
x
(
a
)
,
t
(
a
)
)
=
(
a
,

t
∗

)
,
(
x
(
b
)
,
t
(
b
)
)
=
(
b
,

t
∗

)

(x(a),t(a)) = (a,{t_ \ast }),(x(b),t(b)) = (b,{t_\ast })

, and

t
(
s
)
>

t
∗

t(s) > {t_\ast }

when

a
>
s
>
b

a > s > b

. Let

a
>

x
∗

>
b

a > {x_\ast } > b

. We prove the existence of Gauss interpolation formulas for C and the point

(

x
∗

,

t
∗

)

({x_\ast },{t_\ast })

, for solutions u of the one-dimensional heat equation,

u
t

=

u

x
x

{u_t} = {u_{xx}}

. Such formulas approximate

u
(

x
∗

,

t
∗

)

u({x_\ast },{t_\ast })

in terms of a linear combination of its values on C. The formulas are characterized by the requirement that they are exact for as many basis functions (the heat polynomials) as possible.
Evaluation of the integral ∫^{∞}₀𝑡^{2^{𝛼}-1}𝐽𝜈(𝜒√(1+𝑡²))/(1+𝑡²)^{𝛼+𝛽-1}𝑑𝑡
Authors: Schmidt Paul W. 
Published: June 2010

Methods are developed for evaluating the integral
\[

I
ν

α
β

(
x
)
=

∫
0
∞

t

2
α
−
1

J
ν

(
x

1
+

t
2

)

(
1
+

t
2

)

α
+
β
−
1

d
t
,

I_\nu ^{\alpha \beta }(x) = \int _0^\infty {\frac {{{t^{2\alpha - 1}}{J_\nu }(x\sqrt {1 + {t^2}} )}}{{{{(1 + {t^2})}^{\alpha + \beta - 1}}}}dt,}

\]
where

J
ν

(
t
)

{J_\nu }(t)

is the Bessel function of the first kind and order

ν
\nu

,

α
>
0

\alpha > 0

,

β
>
1

/

4

\beta > 1/4

, and

ν
\nu

is real. Only

I
ν

1

/

2
,
1

(
x
)

I_\nu ^{1/2,1}(x)

and

I
ν

α
,
ν

/

2
+
1
−
α

(
x
)

I_\nu ^{\alpha ,\nu /2 + 1 - \alpha }(x)

are included in previously published tables of integrals of Bessel functions. The integrals

I
1

1

/

2
,
1

/

2

(
x
)

I_1^{1/2,1/2}(x)

and

I
2

1

/

2
,
1

(
x
)

I_2^{1/2,1}(x)

are used in a technique developed by I. S. Fedorova for calculating the diameter distribution of long circular cylinders from small-angle x-ray, light, or neutron scattering data. The

I
ν

α
β

(
x
)

I_\nu ^{\alpha \beta }(x)

are shown to be proportional to a G function. From this result, power series expansions and recurrence relations are developed for use in evaluating the

I
ν

α
β

(
x
)

I_\nu ^{\alpha \beta }(x)

. A convenient expression is obtained for the quantity required in Fedorova’s method for computing diameter distributions.