bluuebunny/crossref_metadata_2025_split huggingface dataset:
The qualitative analysis of a dynamical system modeling the formation of two-layer scales on pure metals
Authors: Baker R. L. 
Published: June 2010

F. Gesmundo and F. Viani have modeled the growth rates of two-oxide scales by the system:
\[

d

q
1

d
t

=
m

K
1

2

q
1

−

m
−
1

m

K
2

2

q
2

,

d

q
2

d
t

=
−
m

K
1

2

q
1

+

K
2

2

q
2

.

\frac {{d{q_1}}}{{dt}} = m\frac {{{K_1}}}{{2{q_1}}} - \frac {{m - 1}}{m} \frac {{{K_2}}}{{2{q_2}}},\qquad \frac {{d{q_2}}}{{dt}} = - m\frac {{{K_1}}}{{2{q_1}}} + \frac {{{K_2}}}{{2{q_2}}}.

\]
We provide a complete qualitative analysis of (1.1) by making use of known results about the general n-dimensional dynamical system:
\[

d

p
i

d
t

=
−

∑

j
=
1

n

a

i
j

p
j

,

p
i

(
t
)
>
0
,

i
=
1
,
…
,
n
.

\frac {{d{p_i}}}{{dt}} = - \sum \limits _{j = 1}^n {\frac {{{a_{ij}}}}{{{p_j}}},\quad {p_i}(t) > 0,\qquad i = 1, \ldots ,n.}

\]
We show that for

m
>
1

m > 1

, the Gesmundo-Viani system admits a unique parabolic solution

q
i

(
t
)
=

c
i

t

,

c
i

>
0

{q_i}(t) = {c_i}\sqrt t ,{c_i} > 0

. This parabolic solution attracts all other solutions. Every solution extends uniquely to a solution on

[
0
,
+
∞
)

[0, + \infty )

, such that the extended solution is eventually monotonically increasing. Finally, the trajectory of any solution coincides with a trajectory of the following linear system:
\[

d

q
1

d
t

=
−

m
−
1

m

K
2

2

q
1

+
m

K
2

2

q
2

,

d

q
2

d
t

=

K
2

2

q
1

+
m

K
1

2

q
2

.

\frac {{d{q_1}}}{{dt}} = - \frac {{m - 1}}{m}{\mkern 1mu} \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_2}}}{2}{q_2},\qquad \frac {{d{q_2}}}{{dt}} = \frac {{{K_2}}}{2}{q_1} + m\frac {{{K_1}}}{2}{q_2}.

\]
Functions on noncompact Lie groups with positive Fourier transforms
Authors: Kawazoe Takeshi 
Published: June 2010

Let G be a homogeneous group with the graded Lie algebra or a noncompact semisimple Lie group with finite center. We define the Fourier transform

f
^

\hat f

of f as a family of operators

f
^

(
π
)
=

∫
G

f
(
x
)
π
(
x
)
d
x
(
π
∈

G
^

)

\hat f(\pi ) = {\smallint _G}f(x)\pi (x)dx(\pi \in \hat G)

, and we say that

f
^

\hat f

is positive if all

f
^

(
π
)

\hat f(\pi )

are positive. Then, we construct an integrable function f on G with positive

f
^

\hat f

and the restriction of f to any ball centered at the origin of G is square-integrable, however, f is not square-integrable on G.
Equivariant, almost homeomorphic maps between 𝑆¹ and 𝑆²
Authors: Soma Teruhiko 
Published: June 2010

Let

Π
\Pi

be a Fuchsian group isomorphic to a non-trivial, closed surface group, and let

M
=

H

3

/

Γ

M = {\mathbb {H}^3}/\Gamma

be a hyperbolic 3-manifold admitting an isomorphism

ρ
:
Π
→
Γ

\rho :\Pi \to \Gamma

. Under certain assumptions, Cannon-Thurston and Minsky showed that there exists a

ρ
\rho

-equivariant, surjective, continuous map

f
:

S
∞
1

→

S
∞
2

f:S_\infty ^1 \to S_\infty ^2

. In this paper, we prove that there exist zero-measure sets

Λ
1

{\Lambda ^1}

in

S
∞
1

S_\infty ^1

and

Λ
2

{\Lambda ^2}

in

S
∞
2

S_\infty ^2

such that the restriction

f

|

S
∞
1

−

Λ
1

:

S
∞
1

−

Λ
1

→

S
∞
2

−

Λ
2

f{|_{S_\infty ^1 - {\Lambda ^1}}}:S_\infty ^1 - {\Lambda ^1} \to S_\infty ^2 - {\Lambda ^2}

is a homeomorphism.
Uniqueness of positive solutions of nonlinear second-order equations
Authors: Dalmasso Robert 
Published: June 2010

In this paper we study the uniqueness question of positive solutions of the two-point boundary value problem:

u
(
t
)
+
f
(

|

t

|

,
u
(
t
)
)
=
0
,
−
R
>
t
>
R
,
u
(
±
R
)
=
0

u(t) + f(|t|,u(t)) = 0, - R > t > R,u( \pm R) = 0

where

R
>
0

R > 0

is fixed and

f
:
[
0
,
R
]
×
[
0
,
∞
)
→

R

f:[0,R] \times [0,\infty ) \to \mathbb {R}

is in

C
1

(
[
0
,
R
]
×
[
0
,
∞
)
)

{C^1}([0,R] \times [0,\infty ))

. A uniqueness result is proved when f satisfies some appropriate conditions. Some examples illustrating our theorem are also given.
Pointwise and norm convergence of a class of biorthogonal expansions
Authors: Benzinger Harold E. 
Published: June 2010

Let{uk(x)},{vk(x)},k=0,±1,…,0⩽x⩽1\{ {u_k}(x)\} ,\{ {v_k}(x)\} ,k = 0, \pm 1, \ldots ,0 \leqslant x \leqslant 1, be sequences of functions inL∞(0,1){L^\infty }(0,1), such that(uk,vj)=δkj({u_k},{v_j}) = {\delta _{kj}}. Letϕk(x)=exp2kπix{\phi _k}(x) = \exp \;2k\pi ix. It is shown that if for a givenp,1>p>∞1 > p > \infty, the sequence{uk}\{ {u_k}\}is complete inLp(0,1){L^p}(0,1), and{vk}\{ {v_k}\}is complete inLq(0,1),pq=p+q{L^q}(0,1),pq = p + q, and if theuk{u_k}’s,vj{v_j}’s are asymptotically related to theϕk{\phi _k}’s, in a sense to be made precise, then{uk}\{ {u_k}\}is a basis forLp(0,1){L^p}(0,1), equivalent to the basis{ϕk}\{ {\phi _k}\}, and for everyfinLp(0,1){L^p}(0,1)a.e. This result is then applied to the eigenfunction expansions of a large class of ordinary differential operators.
Inequalities for polynomials on the unit interval
Authors: Rahman Q. I. , Schmeisser G. 
Published: June 2010

Let

p
n

(
z
)
=

∑

k
=
0

n

a
k

z
k

{p_n}(z) = \sum \nolimits _{k = 0}^n {{a_k}{z^k}}

be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if

p
n

(
z
)

{p_n}(z)

has at most

n
−
1

n - 1

distinct zeros in

(
−
1
,
1
)

( - 1,1)

, then
\[

|

a
n

|

⩽

2

n
−
1

(

cos
⁡

π

4
n

)

2
n

max

−
1
⩽
x
⩽
1

|

p
n

(
x
)

|

,

max

−
1
⩽
x
⩽
1

|

p
′

n

(
x
)

|

⩽

(

n
cos
⁡

π

4
n

)

2

max

−
1
⩽
x
⩽
1

|

p
n

(
x
)

|

.

\begin {array}{*{20}{c}} {|{a_n}| \leqslant {2^{n - 1}}{{\left ( {\cos \frac {\pi }{{4n}}} \right )}^{2n}}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|,} \\ {\max \limits _{ - 1 \leqslant x \leqslant 1} |{{p’}_n}(x)| \leqslant {{\left ( {n\cos \frac {\pi }{{4n}}} \right )}^2}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|.} \\ \end {array}

\]
The Riemann problem in gas dynamics
Authors: Smith Randolph G. 
Published: June 2010

We consider the Riemann problem (R.P.) for the

3

×

3

3\, \times \, 3

system of gas dynamics equations in a single space variable. We assume that the specific internal energy

e
=
e
(
v
,

s
)

e = e(v,\,s)

(s = specific entropy, v = specific volume) satisfies the usual hypotheses,

p
v

>

0
,

p

v
v

>

0
,

p
s

>

0

(
p

=

−

e
v

=

{p_v}\, > \,0,\,{p_{vv}}\, > \,0,\,{p_s}\, > \,0\,(p\, = \, - \,{e_v}\, =

pressure); we also assume some reasonable hypotheses about the asymptotic behavior of e. We call functions e satisfying these hypotheses energy functions Theorem 1. For any initial data

(

U
l

,

U
r

)

(

U
l

=

(

v
l

,

p
l

,

u
l

)

({U_l},\,{U_r})\,({U_l}\, = \,({v_l},\,{p_l},\,{u_l})

,

U
r

=

(

v
r

,

p
r

,

u
r

)

{U_r}\, = \,({v_r},\,{p_r},\,{u_r})

, u = flow velocity), the R. P. has a solution. We introduce two conditions:
\[

(

(I)

)

∂

∂
v

p
(
v
,

e
)
⩽

p
2

2
e

a
m
p
;

(
v
,

e

>

0
)
,

\begin {array}{*{20}{c}}\tag {$\text {(I)}$} {\frac {\partial } {{\partial v}}\,p(v,\,e) \leqslant \frac {{{p^2}}} {{2e}}} & {(v,\,e\, > \,0),} \\ \end {array}

\]

\[

(

(II)

)

∂

∂
v

e
(
v
,

p
)

⩾
−

p
2

a
m
p
;

(
v
,

p

>

0
)
.

\begin {array}{*{20}{c}}\tag {$\text {(II)}$} {\frac {\partial }{{\partial v}}\,e(v,\,p)\, \geqslant - \frac {p} {2}} & {(v,\,p\, > \,0).} \\ \end {array}

\]
Theorem 2. (I) is necessary and sufficient for uniqueness of solutions of the R. P. Nonuniqueness persists under small perturbations of the initial data. (I) is implied by the known condition

(

(
∗
)

)

∂

∂
v

e
(
v
,
p
)
>
0

(
v
,
p
>
0
)
,

\begin{equation}\tag {$(\ast )$} {\frac {\partial } {{\partial v}}e(v,p) > 0} \qquad (v,p > 0), \end{equation}

which holds for all usual gases. (I) implies (II). We construct energy functions e that violate (II), that satisfy (II) but violate (I), and that satisfy (I) but violate (*). In all solutions considered, the shocks satisfy the entropy condition and the Lax shock conditions.
On the zeros of Jacobi polynomials 𝑃_{𝑛}^{(𝛼_{𝑛},𝛽_{𝑛})}(𝑥)
Authors: Moak D. S. , Saff E. B. , Varga R. S. 
Published: June 2010

If

r
n

{r_n}

and

s
n

{s_n}

denote, respectively, the smallest and largest zeros of the Jacobi polynomial

P
n

(

α
n

,

β
n

)

P_n^{({\alpha _n},{\beta _n})}

, where

α
n

>

1

{\alpha _n}\, > \,1

,

β
n

−

1

{\beta _n}\, - \,1

, and if

lim

n
→
∞

α
n

/

(
2
n

+

α
n

+

β
n

+

1
)

=

a

{\lim _{n \to \infty }}\,{\alpha _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,a

and if

lim

n
→
∞

β
n

/

(
2
n

+

α
n

+

β
n

+

1
)

=

b

{\lim _{n \to \infty }}{\beta _n}/(2n\, + \,{\alpha _n}\, + \,{\beta _n}\, + \,1)\, = \,b

, then the numbers

r

a
,
b

{r_{a,b}}

and

s

a
,
b

{s_{a,b}}

are determined where
\[

lim

n
→
∞

r

n

=

r

a
,
b

,

lim

n
→
∞

s

n

=

s

a
,
b

\lim \limits _{n \to \infty } \,{r_{n\,}}\, = \,{r_{a,b}},\lim \limits _{n \to \infty } \,{s_{n\,}}\, = \,{s_{a,b}}

\]
. Furthermore, the zeros of

{

P
n

(

α
n

,

β
n

)

(
x
)

}

n
=
0

∞

\{ P_n^{({\alpha _n},{\beta _n})}(x)\} _{n = 0}^\infty

are dense in

[

r

a
,
b

,

s

a
,
b

]

[{r_{a,b}},{s_{a,b}}]

.
On the ranges of analytic functions
Authors: Hwang J. S. 
Published: June 2010

Following Doob, we say that a function

f
(
z
)

f(z)

analytic in the unit disk U has the property

K
(
ρ
)

K(\rho )\,

if

f
(
0
)

=

0

f(0)\, = \,0

and for some

arc

A

\operatorname {arc} \,A\,

on the unit circle whose measure

|
A
|

⩾

2
ρ

>

0

\left | A \right |\, \geqslant \,2\rho \, > \,0

,
\[

lim

inf

i
→
∞

|

f
(

P
i

)

|

⩾

1

where

P
i

→

P

∈

A

and

P
i

∈

U
.

\lim \,\inf \limits _{i \to \infty } \,\left | {f({P_i})} \right |\, \geqslant \,1\,{\text {where}}\,{P_i}\, \to \,P\, \in \,A\,{\text {and}}\,{P_i}\, \in \,U.

\]
We recently have solved a problem of Doob by showing that there is an integer

N
(
ρ
)

N(\rho )

such that no function with the property

K
(
ρ
)

K(\rho )

can satisfy
\[

(
1

−

|
z
|

)

|

f
n

′

(
z
)

|

⩽

1

/

n

for

z

∈

U
,

where

n

>

N
(
ρ
)
.

(1\, - \,\left | z \right |)\left | {{f_n}’ (z)} \right |\, \leqslant \,1/n\,{\text {for}}\,z\, \in \,U,\,{\text {where}}\,n\, > \,N(\rho ).

\]
The function
\[

f
n

(
z
)

=

1

+

(
1

−

z
n

)

/

n
2

,

{f_n}(z)\, = \,1\, + \,(1\, - \,{z^n})/{n^2},

\]
shows that the condition

f
n

(
0
)

=

0

{f_n}(0)\, = \,0

is necessary and cannot be replaced by

f
n

(
0
)

=

r

e

i
α

{f_n}(0)\, = \,r{e^{i\alpha }}

, for

r

>

1

r\, > \,1

. Naturally, we may ask whether this can be replaced by

f
n

(
0
)

=

r

e

i
α

{f_n}(0)\, = \,r{e^{i\alpha }}

, for

r

>

1

r\, > \,1

? The answer turns out to be yes, when

n

>

N

(
r
,

ρ
)

n\, > \,N\,(r,\,\rho )

, where
\[

N
(
r
,

ρ
)

≑

(
1

/

(
1

−

r
)
)
log
⁡
(
1

/

(
1

−

cos
⁡
ρ
)
)
.

N(r,\,\rho )\,\doteqdot \,(1/(1\, - \,r))\log (1/(1\, - \,\cos \rho )).

\]
.
Uniqueness of invariant means for measure-preserving transformations
Authors: Rosenblatt Joseph 
Published: June 2010

For some compact abelian groups

X
X

(e.g.

T
n

T^n

,

n
⩾
2

n \geqslant 2

, and

∏

n
=
1

∞

Z
2

\prod \nolimits _{n = 1}^\infty {{Z_2}}

), the group

G
G

of topological automorphisms of

X
X

has the Haar integral as the unique

G
G

-invariant mean on

L
∞

(
X
,

λ
X

)

{L_\infty }(X,{\lambda _X})

. This gives a new characterization of Lebesgue measure on the bounded Lebesgue measurable subsets

β
\beta

of

R
n

{R^n}

,

n
⩾
3

n \geqslant 3

; it is the unique normalized positive finitely-additive measure on

β
\beta

which is invariant under isometries and the transformation of

R
n

:
(

x
1

,
…
,

x
n

)
↦
(

x
1

+

x
2

,

x
2

,
…
,

x
n

)

{R^n}:({x_1}, \ldots ,{x_n}) \mapsto ({x_1} + {x_2},{x_2}, \ldots ,{x_n})

. Other examples of, as well as necessary and sufficient conditions for, the uniqueness of a mean on

L
∞

(
X
,
β
,
p
)

{L_\infty }(X,\beta ,p)

, which is invariant by some group of measure-preserving transformations of the probability space

(
X
,
β
,
p
)

(X,\beta ,p)

, are described.
Singular integrals and maximal functions associated with highly monotone curves
Authors: Nestlerode W. C. 
Published: June 2010

Let

γ
:
[
−
1
,
1
]
→

R

n

\gamma :[ - 1,1] \to {{\mathbf {R}}^n}

be an odd curve. Set
\[

H
γ

f
(
x
)
=

PV

∫

f
(
x
−
γ
(
t
)
)

(
d
t

/

t
)

{H_\gamma }f(x) = {\text {PV}}\int {f(x - \gamma (t))\,(dt/t)}

\]
and
\[

M
γ

f
(
x
)
=
sup

h

−
1

∫
0
h

|

f
(
x
−
γ
(
t
)
)

|

d
t

{M_\gamma }f(x) = \sup {h^{ - 1}}\int _0^h {|f(x - \gamma (t))|\,dt}

\]
. We introduce a class of highly monotone curves in

R

n

{{\mathbf {R}}^n}

,

n
⩾
2

n \geqslant 2

, for which we prove that

H
γ

{H_\gamma }

and

M
γ

{M_\gamma }

are bounded operators on

L
2

(

R

n

)

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

. These results are known if

γ
\gamma

has nonzero curvature at the origin, but there are highly monotone curves which have no curvature at the origin. Related to this problem, we prove a generalization of van der Corput’s estimate of trigonometric integrals.
Spectral permanence for joint spectra
Authors: Curto Raul E. 
Published: June 2010

For a

C

∗

{C^{\ast }}

-subalgebra

A
A

of a

C

∗

{C^{\ast }}

-algebra

B
B

and a commuting

n
n

-tuple

a
=
(

a
1

,
…
,

a
n

)

a = ({a_1}, \ldots ,{a_n})

of elements of

A
A

, we prove that

Sp
⁡
(
a
,

A
)
=
Sp
⁡
(
a
,

B
)

\operatorname {Sp} (a,\,A) = \operatorname {Sp} (a,\,B)

, where

Sp
\operatorname {Sp}

denotes Taylor spectrum. As a consequence we prove that

0
∉
Sp
⁡
(
a
,

A
)

0 \notin \operatorname {Sp} (a,\,A)

if and only if
\[

a
^

=

(

d
1

a
m
p
;

a
m
p
;

d
2

∗

a
m
p
;

d
3

a
m
p
;

a
m
p
;

d
4

∗

a
m
p
;
⋱

)

∈
L

(

A
⊗

C

2

n
−
1

)

\hat a = \left ( {\begin {array}{*{20}{c}} {{d_1}} & {} & {} \\ {d_2^{\ast }} & {{d_3}} & {} \\ {} & {d_4^{\ast }} & \ddots \\ \end {array} } \right ) \in L\left ( {A \otimes {{\mathbf {C}}^{{2^{n - 1}}}}} \right )

\]
is invertible, where

d
i

{d_i}

is the

i
i

th boundary map in the Koszul complex for

A
A

. More generally, we show that

σ

δ
,
k

(
a
,

A
)
=

σ

δ
,
k

(

a
,

B

)

{\sigma _{\delta ,k}}(a,\,A) = {\sigma _{\delta ,k}}\left ( {a,\,B} \right )

and

σ

π
,
k

(
a
,

A
)
=

σ

π
,
k

(
a
,

B
)

{\sigma _{\pi ,k}}(a,\,A) = {\sigma _{\pi ,k}}(a,\,B)

(all

k
k

), where

σ

δ
,
⋅

{\sigma _{\delta ,\cdot }}

and

σ

π
,
⋅

{\sigma _{\pi ,\cdot }}

are the joint spectra considered by Z. Słodkowski.
Tauberian conditions for 𝐿¹-convergence of Fourier series
Authors: Stanojević Časlav V. 
Published: June 2010

It is proved that Fourier series with asymptotically even coefficients and satisfying

lim

λ
→
1

lim

sup

n
→
∞

∑

j
=
n

[
λ
n
]

j

p
−
1

|

Δ

f
^

(
j
)

|

p

=
0

{\lim _{\lambda \to 1}}\lim {\sup _{n \to \infty }}\sum _{j = n}^{[\lambda n]}{j^{p - 1}}|\Delta \hat f(j){|^p} = 0

, for some

1
>
p
⩽
2

1 > p \leqslant 2

, converge in

L
1

{L^1}

-norm if and only if

|

|

f
^

(
n
)

E
n

+

f
^

(
−
n
)

E

−
n

|

|

=
o
(
1
)

||\hat f(n){E_n} + \hat f( - n){E_{ - n}}|| = o(1)

, where

E
n

(
t
)
=

∑

k
=
0

n

e

i
k
t

{E_n}(t) = \sum _{k = 0}^n{e^{ikt}}

. Recent results of Stanojević [1], Bojanic and Stanojević [2], and Goldberg and Stanojević [3] are special cases of some corollaries to the main theorem.
Nonstandard construction of the stochastic integral and applications to stochastic differential equations. II
Authors: Hoover Douglas N. , Perkins Edwin 
Published: June 2010

H. J. Keisler has recently used a nonstandard theory of Itô integration (due to R. M. Anderson) to construct solutions of Itô integral equations by solving an associated internal difference equation. In this paper we use the same general approach to find solutions

y
(
t
)

y(t)

of semimartingale integral equations of the form
\[

y
(
t
,
ω
)
=
h
(
t
,
ω
)
+

∫
0
t

f
(
s
,
ω
,
y
(
⋅
,
ω
)
)

d
z
(
s
)

y(t,\omega ) = h(t,\omega ) + \int _0^t {f(s,\omega ,y(\cdot ,\omega ))\,dz(s)}

\]
, where

z
z

is a given semimartingale,

h
h

is a right-continuous process and

f
(
s
,
ω
,
⋅
)

f(s,\omega , \cdot )

is continuous on the space of right-continuous functions with left limits, with the topology of uniform convergence on compacts. In addition, we generalize Keisler’s continuity theorem and give necessary and sufficient conditions for an internal martingale to be

S
S

-continuous.
Tauberian 𝐿¹-convergence classes of Fourier series. I
Authors: Bray William O. , Stanojević Časlav V. 
Published: June 2010

It is shown that the Stanojević [2] necessary and sufficient conditions for

L
1

{L^1}

-convergence of Fourier series of

f
∈

L
1

(
T
)

f \in {L^1}(T)

can be reduced to the classical form. A number of corollaries of a recent Tauberian theorem are obtained for the subclasses of the class of Fourier coefficients satisfying

n
α

|

Δ

f
^

(
n
)

|

=
o
(
l
)

(
n
→
∞
)

{n^\alpha }|\Delta \hat {f}(n)| = o(l)\,(n \to \infty )

for some

0
>
α
⩽

1
2

0 > \alpha \leqslant \frac {1}{2}

. For Fourier series with coefficients asymptotically even with respect to a sequence

{

l
n

}
,

l
n

=
o
(
n
)

(
n
→
∞
)

\{{l_n}\} ,{l_n} = o(n)\,(n \to \infty )

, and satisfying
\[

l
n

−
1

/

q

(

∑

k
=
n

n
+
[
n

/

l
n

]

k

p
−
1

|

Δ

f
^

(
k
)

|

p

)

1

/

p

=
o
(
1
)

(
n
→
∞
)
,

1

/

p
+
1

/

q
=
1
,

l_n^{ - 1/q}{\left ({\sum \limits _{k = n}^{n + [n/{l_n}]} {{k^{p - 1}}|\Delta \,\hat f(k)} {|^p}} \right )^{1/p}} = o(1)\, \quad (n \to \infty ), \quad 1/p + 1/q = 1,

\]
necessary and sufficient conditions for

L
1

{L^1}

-convergence are obtained. In particular for

l
n

=
[
∥

σ
n

(
f
)
−
f

∥

−
1

]

{l_n} = [\parallel {\sigma _n}(f) - f{\parallel ^{ - 1}}]

, an important corollary is obtained which connects smoothness of

f
f

with smoothness of

{

f
^

(
n
)
}

\{\hat f(n)\}

.
A sphere theorem for manifolds of positive Ricci curvature
Authors: Shiohama Katsuhiro 
Published: June 2010

Instead of injectivity radius, the contractibility radius is estimated for a class of complete manifolds such that

Ri

c

M

⩾
1
,

K
M

⩾
−

κ
2

{\text {Ri}}{{\text {c}}_M} \geqslant 1,{K_M} \geqslant - {\kappa ^2}

and the volume of

M
⩾

M \geqslant

the volume of the

(
π
−
ε
)

(\pi - \varepsilon )

-ball on the unit

m
m

-sphere,

m
=

dim

M

m = {\text {dim }}M

. Then for a suitable choice of

ε
=
ε
(
m
,
k
)

\varepsilon = \varepsilon (m,k)

every

M
M

belonging to this class is homeomorphic to

S
m

{S^m}

.
Decay of Walsh series and dyadic differentiation
Authors: Wade William R. 
Published: June 2010

Let

W
2

n

[
f
]

{W_2}\,n\,[f]

denote the

2
n

th

{2^n}{\text {th}}

partial sums of the Walsh-Fourier series of an integrable function

f
f

. Let

ρ
n

(
x
)

{\rho _n}(x)

represent the ratio

W
2

n
[
f
,
x
]

/

2
n

{W_2}n[f,x]/{2^n}

, for

x
∈
[
0
,
1
]

x \in [0,1]

, and let

T
(
f
)

T(f)

represent the function

(
Σ

ρ
n
2

)

1

/

2

{(\Sigma \rho _n^2)^{1/2}}

. We prove that

T
(
f
)

T(f)

belongs to

L
p

[
0
,
1
]

{L^p}[0,1]

for all

0
>
p
>
∞

0 > p > \infty

. We observe, using inequalities of Paley and Sunouchi, that the operator

f
→
T
(
f
)

f \to T(f)

arises naturally in connection with dyadic differentiation. Namely, if

f
f

is strongly dyadically differentiable (with derivative

D
˙

f

\dot Df

) and has average zero on the interval [0, 1], then the

L
p

{L^p}

norms of

f
f

and

T
(

D
˙

f
)

T(\dot Df)

are equivalent when

1
>
p
>
∞

1 > p > \infty

. We improve inequalities implicit in Sunouchi’s work for the case

p
=
1

p = 1

and indicate how they can be used to estimate the

L
1

{L^1}

norm of

T
(

D
˙

f
)

T(\dot Df)

and the dyadic

H
1

{H^1}

norm of

f
f

by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if

f
f

is strongly dyadically differentiable in dyadic

H
1

{H^1}

, then

∫
0
1

Σ

N
=
1

∞

|

W
N

[
f
,
x
]
−

σ
N

[
f
,
x
]

/

N

d
x
>
∞

\int _0^1 {\Sigma _{N = 1}^\infty |{W_N}[f,x] - {\sigma _N}[f,x]/N\,dx > \infty }

.
An improved stability result for resonances
Authors: Ashbaugh Mark S. , Sundberg Carl 
Published: June 2010

We prove stability of shape resonances for the sequence of Schrödinger equations

(
−

d
2

/

d

x
2

+
U
(
x
)
+

W
n

(
x
)
)
ψ
(
x
)
=
E
ψ
(
x
)
,
0
⩽
x

>
∞

( - {d^2}/d{x^2} + U(x) + {W_n}(x))\psi (x) = E\psi (x),0 \leqslant x\, > \infty

, in the limit

n
→
∞

n \to \infty

where the barrier potentials

W
n

(
x
)

{W_n}(x)

are integrable, nonnegative, supported in the interval

[
1
,
a
]

(
1
>
a
>
∞
)

[1,a]\;(1 > a > \infty )

, and approach infinity pointwise a.e. for

x
∈
[
1
,
a
]

x \in [1,a]

as

n
→
∞

n \to \infty

. In the course of our investigation we prove that for suitable complex initial conditions the solution to the Riccati equation

S
′

(
x
)
=
1
−
(

W
n

(
x
)
−
E
)

[
S
(
x
)

]
2

S’(x) = 1 - ({W_n}(x) - E){[S(x)]^2}

goes to

0
0

as

n
→
∞

n \to \infty

uniformly on compact subsets of

[
1
,
a
]

[1,a]

. Our approach is via ordinary differential equations using outgoing wave boundary conditions to define resonances. Our stability result extends a similar result of Ashbaugh and Harrell, who use an argument based on asymptotics and the implicit function theorem to study the above problem with

λ
V
(
x
)

\lambda V(x)

replacing

W
n

(
x
)

{W_n}(x)

. Our approach is to use the Riccati equation analysis mentioned above and an application of Hurwitz’s Theorem from complex variable theory.
The null space and the range of a convolution operator in a fading memory space
Authors: Staffans Olof J. 
Published: June 2010

We study the convolution equation

(
∗
)

(\ast )

\[

μ

∗

x
′

(
t
)
+
v

∗

x
(
t
)
=
f
(
t
)

(
−
∞
>
t

>
∞
)

\mu \; \ast \;x’(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty > t\, > \infty )

\]
, as well as a perturbed version of

(
∗
)

(\ast )

, namely

(
∗
∗
)

(\ast \ast )

\[

μ

∗

x
′

(
t
)
+
v

∗

x
(
t
)
=

F
(
x
)

(
t
)

(
−
∞
>
t
>
∞
)
.

\mu \;\ast \;x’(t) + v\,\ast \;x(t) = \,F(x)\,(t)\quad ( - \infty > t > \infty ).

\]
Here

x
x

is a

R

n

{{\mathbf {R}}^n}

-valued function on

(
−
∞
,
∞
)
,

x
′

(
t
)
=
d
x
(
t
)

/

d
t

( - \infty ,\infty ),x’(t) = dx(t)/dt

, and

μ
\mu

and

ν
\nu

are matrix-valued measures. If

μ
\mu

and

ν
\nu

are supported on

[
0
,
∞
)

[0,\infty )

, with

μ
\mu

atomic at zero, then

(
∗
)

(\ast )

can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that

μ
\mu

and

ν
\nu

are supported on

[
0
,
∞
)

[0,\infty )

, prescribe an initial condition of the type

x
(
t
)
=
ξ
(
t
)

(
t
⩽
0
)

x(t) = \xi (t)\,(t \leqslant 0)

, and require

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

to hold only for

t
⩾
0

t \geqslant 0

. Instead we permit

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

to be of "Fredholm" type, i.e.

μ
\mu

and

ν
\nu

need not vanish on

(
−
∞
,
0
)

( - \infty ,0)

, we restrict the growth rate of

x
x

and

f
f

at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of

(
∗
)

(\ast )

and

(
∗
∗
)

(\ast \ast )

on the whole real line, satisfying conditions like

|

x
(
t
)

|

⩽
C
η
(
t
)

(
−
∞
>
t
>
∞
)

|x(t)| \leqslant C\eta (t)\;( - \infty > t > \infty )

, where

C
C

is a constant, depending on

x
x

, and

η
\eta

is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of

(
∗
)

(\ast )

has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.