bluuebunny/crossref_metadata_2025_split huggingface dataset:
On existence and uniqueness for a new class of nonlinear partial differential equations using compactness methods and differential difference schemes
Authors: Dushane Theodore E. 
Published: July 2010

We prove existence and uniqueness results for the following Cauchy problem in the half plane

t
≥
0
:

u
t

+

(
f
(
u
)

)
x

+

u

x
x
x

=

g
1

(
u
)

u

x
x

+

g
2

(
u
)

(

u
x

)
2

+
p
(
u
)
,
u
(
x
,
0
)
=

u
0

(
x
)

t \geq 0:{u_t} + {(f(u))_x} + {u_{xxx}} = {g_1}(u){u_{xx}} + {g_2}(u){({u_x})^2} + p(u),u(x,0) = {u_0}(x)

, where

u
=
u
(
x
,
t
)

u = u(x,t)

and the subscripts indicate partial derivatives. We require that f,

g
1

{g_1}

,

g
2

{g_2}

, and p be sufficiently smooth and satisfy

f
′

(
u
)
≥
0
,

∫
0
u

f
(
v
)

d
v
≥
0

f’(u) \geq 0,\smallint _0^uf(v)\;dv \geq 0

, and other similar sign conditions on

g
1

{g_1}

,

g
2

{g_2}

, and p. Our hypotheses allow for exponential growth of f,

g
1

{g_1}

,

g
2

{g_2}

, and p so long as the sign conditions are satisfied and include the special cases

f
(
u
)
=

u

2
n
+
1

,

g
1

(
u
)
=

u

2
m

,

g
2

(
u
)
=
−

u

2
r
+
1

f(u) = {u^{2n + 1}},{g_1}(u) = {u^{2m}},{g_2}(u) = - {u^{2r + 1}}

, and

p
(
u
)
=
−

u

2
s
+
1

p(u) = - {u^{2s + 1}}

, for n, m, r, and s nonnegative integers. To obtain a global solution in time, we perturb the equation by

−
ϵ
(

u

x
x
x
x

−

(
f
(
u
)

)

x
x

)

- \epsilon ({u_{xxxx}} - {(f(u))_{xx}})

. The perturbed equation is solved locally (in time) and this solution is extended to a global solution by means of a priori estimates on the

H
s

{H^s}

(of space) norms of the local solution. These estimates require the use of new nonlinear functionals. We then obtain the solution to the original equation as a limit of solutions to the perturbed equation as

ϵ
\epsilon

tends to zero using the standard techniques. For the related periodic problem, for which we require

u
(
x
+
2
π
,
t
)
=
u
(
x
,
t
)

u(x + 2\pi ,t) = u(x,t)

for all

t
≥
0

t \geq 0

, we also obtain existence and uniqueness results. We prove existence for this problem via similar techniques to the nonperiodic case. We then consider differential difference schemes for the periodic initial value problem and show that we may obtain the solution as the limit of solutions to an appropriate scheme.
Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+ℱ(𝓊)
Authors: Levine Howard A. 
Published: July 2010

For the equation in the title, let P and A be positive semidefinite operators (with P strictly positive) defined on a dense subdomain

D
⊆
H

D \subseteq H

, a Hilbert space. Let D be equipped with a Hilbert space norm and let the imbedding be continuous. Let

F

:
D
→
H

\mathcal {F}:D \to H

be a continuously differentiable gradient operator with associated potential function

G

\mathcal {G}

. Assume that

(
x
,

F

(
x
)
)
≥
2
(
2
α
+
1
)

G

(
x
)

(x,\mathcal {F}(x)) \geq 2(2\alpha + 1)\mathcal {G}(x)

for all

x
∈
D

x \in D

and some

α
>
0

\alpha > 0

. Let

E
(
0
)
=

1
2

[
(

u
0

,
A

u
0

)
+
(

v
0

,
P

v
0

)
]

E(0) = \tfrac {1}{2}[({u_0},A{u_0}) + ({v_0},P{v_0})]

where

u
0

=
u
(
0
)
,

v
0

=

u
t

(
0
)

{u_0} = u(0),{v_0} = {u_t}(0)

and

u
:
[
0
,
T
)
→
D

u:[0,T) \to D

be a solution to the equation in the title. The following statements hold: If

G

(

u
0

)
>
E
(
0
)

\mathcal {G}({u_0}) > E(0)

, then

lim

t
→

T
−

(
u
,
P
u
)
=
+
∞

{\lim _{t \to {T^ - }}}(u,Pu) = + \infty

for some

T
>
∞

T > \infty

. If

(

u
0

,
P

v
0

)
>
0
,
0
>
E
(
0
)
−

G

(

u
0

)
>
α

(

u
0

,
P

v
0

)
2

/

4
(
2
α
+
1
)
(

u
0

,
P

u
0

)

({u_0},P{v_0}) > 0,0 > E(0) - \mathcal {G}({u_0}) > \alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0})

and if u exists on

[
0
,
∞
)

[0,\infty )

, then (u,Pu) grows at least exponentially. If

(

u
0

,
P

v
0

)
>
0

({u_0},P{v_0}) > 0

and

α

(

u
0

,
P

v
0

)
2

/

4
(
2
α
+
1
)
(

u
0

,
P

u
0

)
≤
E
(
0
)
−

G

(

u
0

)
>

1
2

(

u
0

,
P

v
0

)
2

/

(

u
0

,
P

u
0

)

\alpha {({u_0},P{v_0})^2}/4(2\alpha + 1)({u_0},P{u_0}) \leq E(0) - \mathcal {G}({u_0}) > \tfrac {1}{2}{({u_0},P{v_0})^2}/({u_0},P{u_0})

and if the solution exists on

[
0
,
∞
)

[0,\infty )

, then (u,Pu) grows at least as fast as

t
2

{t^2}

. A number of examples are given.
Conjugate points, triangular matrices, and Riccati equations
Authors: Nehari Zeev 
Published: July 2010

Let

A
A

be a real continuous

n
×
n

n \times n

matrix on an interval

Γ
\Gamma

, and let the

n
n

-vector

x
x

be a solution of the differential equation

x
′

=
A
x

x’ = Ax

on

Γ
\Gamma

. If

[
α
,
β
]
∈
Γ
,
β

[\alpha ,\beta ] \in \Gamma ,\beta

is called a conjugate point of

α
\alpha

if the equation has a nontrivial solution vector

x
=
(

x
1

,
…
,

x
n

)

x = ({x_1}, \ldots ,{x_n})

such that

x
1

(
α
)
=
…
=

x
k

(
α
)
=

x

k
+
1

(
β
)
=
…
=

x
n

(
β
)
=
0

{x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0

for some

k
∈
[
1
,
n
−
1
]

k \in [1,n - 1]

. It is shown that the absence on

(

t
1

,

t
2

)

({t_1},{t_2})

of a point conjugate to

t
1

{t_1}

with respect to the equation

x
′

=
A
x

x’ = Ax

is equivalent to the existence on

(

t
1

,

t
2

)

({t_1},{t_2})

of a continuous matrix solution

L
L

of the nonlinear differential equation

L
′

=

[
L

A
∗

L

−
1

]

τ
0

L

L’ = {[L{A^ \ast }{L^{ - 1}}]_{{\tau _0}}}L

with the initial condition

L
(

t
1

)
=
I

L({t_1}) = I

, where

[
B

]

τ
0

{[B]_{{\tau _0}}}

denotes the matrix obtained from the

n
×
n

n \times n

matrix

B
B

by replacing the elements on and above the main diagonal by zeros. This nonlinear equation—which may be regarded as a generalization of the Riccati equation, to which it reduces for

n
=
2

n = 2

—can be used to derive criteria for the presence or absence of conjugate points on a given interval.
Limit theorems for variational sums
Authors: Hudson William N. , Tucker Howard G. 
Published: July 2010

Limit theorems in the sense of a.s. convergence, convergence in

L
1

{L_1}

-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if

{
X
(
t
)
,
0
≤
t
≤
T
}

\{ X(t),0 \leq t \leq T\}

is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form

Σ

k
=
1

n

g
(
X
(

t

n
k

)
−
X
(

t

n
,
k
−
1

)
)

\Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))

is shown to converge with probability one and in

L
1

{L_1}

-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.
Modular constructions for combinatorial geometries
Authors: Brylawski Tom 
Published: July 2010

R. Stanley, in an investigation of modular flats in geometries (Algebra Universalis 1-2 (1971), 214-217), proved that the characteristic polynomial

χ
(
x
)

\chi (x)

of a modular flat

x
x

divides the characteristic polynomial

χ
(
G
)

\chi (G)

of a geometry

G
G

. In this paper we identify the quotient: THEOREM. If

x
x

is a modular flat of

G
,
χ
(
G
)

/

χ
(
x
)
=
χ
(

T
x

¯

(
G
)
)

/

(
λ
−
1
)

G,\chi (G)/\chi (x) = \chi (\overline {{T_x}} (G))/(\lambda - 1)

, where

T
x

¯

(
G
)

\overline {{T_x}} (G)

is the complete Brown truncation of

G
G

by

x
x

. (The lattice of

T
x

¯

(
G
)

\overline {{T_x}} (G)

consists of all flats containing

x
x

and all flats disjoint from

x
x

, with the induced order from

G
G

.) We give many characterizations of modular flats in terms of their lattice properties as well as by means of a short-circuit axiom and a modular version of the MacLane-Steinitz exchange axiom. Modular flats are shown to have many of the useful properties of points and distributive flats (separators) in addition to being much more prevalent. The theorem relating the chromatic polynomials of two graphs and the polynomial of their vertex join across a common clique generalizes to geometries: THEOREM. Given geometries

G
G

and

H
H

, if

x
x

is a modular flat of

G
G

as well as a subgeometry of

H
H

, then there exists a geometry

P
=

P
x

(
G
,
H
)

P = {P_x}(G,H)

which is a pushout in the category of injective strong maps and such that

χ
(
P
)
=
χ
(
G
)
χ
(
H
)

/

χ
(
x
)

\chi (P) = \chi (G)\chi (H)/\chi (x)

. The closed set structure, rank function, independent sets, and lattice properties of

P
P

are characterized. After proving a modular extension theorem we give applications of our results to Crapo’s single element extension theorem, Crapo’s join operation, chain groups, unimodular geometries, transversal geometries, and graphs.
On the dimension of varieties of special divisors
Authors: Lax R. F. 
Published: July 2010

Let

T
g

{T_g}

denote the Teichmüller space and let

V
V

denote the universal family of Teichmüller surfaces of genus

g
g

Let

V

T
g

(
n
)

V_{{T_g}}^{(n)}

denote the

n
n

th symmetric product of

V
V

over

T
g

{T_g}

and let

J
J

denote the family of Jacobians over

T
g

{T_g}

. Let

f
:

V

T
g

(
n
)

→
J

f:V_{{T_g}}^{(n)} \to \text {J}

be the natural relativization over

T
g

{T_g}

of the classical map defined by integrating holomorphic differentials. Let
\[

u
:

f
∗

Ω

J

/

T
g

1

→

Ω

V

T
g

/

T
g

(
n
)

1

u:{f^\ast }\Omega _{\text {J} /{T_g}}^1 \to \Omega _{V_{{T_g}/{T_g}}^{(n)}}^1

\]
be the map induced by

f
f

. We define

G
n
r

G_n^r

to be the analytic subspace of

V

T
g

(
n
)

V_{{T_g}}^{(n)}

defined by the vanishing of

∧

n
−
r
+
1

u

{ \wedge ^{n - r + 1}}u

. Put

τ
=
(
r
+
1
)
(
n
−
r
)
−
r
g

\tau = (r + 1)(n - r) - rg

. We show that

G
n
1

−

G
n
2

G_n^1 - G_n^2

, if nonempty, is smooth of pure dimension

3
g
−
3
+
τ
+
1

3g - 3 + \tau + 1

. From this result, we may conclude that, for a generic curve

X
X

, the fiber of

G
n
1

−

G
n
2

G_n^1 - G_n^2

over the module point of

X
X

, if nonempty, is smooth of pure dimension

τ
+
1

\tau + 1

, a classical assertion. Variational formulas due to Schiffer and Spencer and Rauch are employed in the study of

G
n
r

G_n^r

.
Generalization of right alternative rings
Authors: Hentzel Irvin Roy , Piacentini Cattaneo Giulia Maria 
Published: July 2010

We study nonassociative rings

R
R

satisfying the conditions (1)

(
a
b
,
c
,
d
)
+
(
a
,
b
,
[
c
,
d
]
)
=
a
(
b
,
c
,
d
)
+
(
a
,
c
,
d
)
b

(ab,c,d) + (a,b,[c,d]) = a(b,c,d) + (a,c,d)b

for all

a
,
b
,
c
,
d
∈
R

a,b,c,d \in R

, and (2)

(
x
,
x
,
x
)
=
0

(x,x,x) = 0

for all

x
∈
R

x \in R

. We furthermore assume weakly characteristic not 2 and weakly characteristic not 3. As both (1) and (2) are consequences of the right alternative law, our rings are generalizations of right alternative rings. We show that rings satisfying (1) and (2) which are simple and have an idempotent

≠
0
,
≠
1

\ne 0, \ne 1

, are right alternative rings. We show by example that

(
x
,
e
,
e
)

(x,e,e)

may be nonzero. In general,

R
=

R
′

+
(
R
,
e
,
e
)

R = R’ + (R,e,e)

(additive direct sum) where

R
′

R’

is a subring and

(
R
,
e
,
e
)

(R,e,e)

is a nilpotent ideal which commutes elementwise with

R
R

. We examine

R
′

R’

under the added assumption of Lie admissibility: (3)

(
a
,
b
,
c
)
+
(
b
,
c
,
a
)
+
(
c
,
a
,
b
)
=
0

(a,b,c) + (b,c,a) + (c,a,b) = 0

for all

a
,
b
,
c
∈
R

a,b,c \in R

. We generate the Peirce decomposition. If

R
′

R’

has no trivial ideals contained in its center, the table for the multiplication of the summands is associative, and the nucleus of

R
′

R’

contains

R

10

′

+

R

01

′

{R’_{10}} + {R’_{01}}

. Without the assumption on ideals, the table for the multiplication need not be associative; however, if the multiplication is defined in the most obvious way to force an associative table, the new ring will still satisfy (1), (2), (3).
On Dedekind’s problem: the number of isotone Boolean functions. II
Authors: Kleitman D. , Markowsky G. 
Published: July 2010

It is shown that

ψ
(
n
)

\psi (n)

, the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies
\[

ψ
(
n
)
⩽

2

(
1
+
O
(
log

n

/

n
)
)

(

n

[
n

/

2
]

)

.

\psi (n) \leqslant {2^{(1 + O(\log \;n/n))\left ( {\begin {array}{*{20}{c}} n \\ {[n/2]} \\ \end {array} } \right )}}.

\]
This result is an improvement by a factor

n

\sqrt n

in the 0 term of a previous result of Kleitman. In the course of deriving the main result, we analyze thoroughly the techniques used here and earlier by Kleitman, and show that the result in this paper is “best possible” (up to constant) using these techniques.
Measures associated with Toeplitz matrices generated by the Laurent expansion of rational functions
Authors: Day K. Michael 
Published: July 2010

Let

T
n

(
a
)
=
(

a

i
−
j

)

i
,
j
=
0

n

{T_n}(a) = ({a_{i - j}})_{i,j = 0}^n

be the finite Toeplitz matrices generated by the Laurent expansion of an arbitrary rational function, and let

σ
n

=
{

λ

n
0

,
…
,

λ

n
n

}

{\sigma _n} = \{ {\lambda _{n0}}, \ldots ,{\lambda _{nn}}\}

be the corresponding sets of eigenvalues of

T
n

(
f
)

{T_n}(f)

. Define a sequence of measures

α
n

,

α
n

(
E
)
=

(
n
+
1

)

−
1

Σ

λ

n
i

∈
E

1
,

λ

n
i

∈

σ
n

{\alpha _n},{\alpha _n}(E) = {(n + 1)^{ - 1}}{\Sigma _{{\lambda _{ni}} \in E}}1,{\lambda _{ni}} \in {\sigma _n}

, and

E
E

a set in the

λ
\lambda

-plane. It is shown that the weak limit

α
\alpha

of the measures

α
n

{\alpha _n}

is unique and possesses at most two atoms, and the function

f
f

which give rise to atoms are identified.
Hypercontractive semigroups and Sobolev’s inequality
Authors: Feissner George F. 
Published: July 2010

If

H
⩾
0

H \geqslant 0

is the generator of a hypercontractive semigroup (HCSG), it is known that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is a bounded operator from

L
p

{L^p}

to

L
p

,
1
⩽
p
⩽
∞

{L^p},1 \leqslant p \leqslant \infty

. We prove that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
2

{L^2}

to the Orlicz space

L
2

I

n

+

L

{L^2}{\text { I}}{{\text {n}}^ + }L

, basing the proof on the uniform semiboundedness of the operator

H
+
V

H + V

, for suitable

V
V

. We also prove by an interpolation argument, that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
p

{L^p}

to

L
p

I

n

+

L
,
2
⩽
p
>
∞

{L^p}{\text { I}}{{\text {n}}^ + }L,2 \leqslant p > \infty

. Another interpolation argument shows that

(
H
+
1

)

−
1

/

2

{(H + 1)^{ - 1/2}}

is bounded from

L
p

(

I

n

+

L

)
m

{L^p}{({\text {I}}{{\text {n}}^ + }L)^m}

to

L
p

(

I

n

+

L

)

m
+
1

{L^p}{({\text {I}}{{\text {n}}^ + }L)^{m + 1}}

and

m
m

a positive integer. Finally, we identify the topological duals of the spaces mentioned above.
Some inequalities for polynomials with a prescribed zero
Authors: Rahman Q. I. , Schmeisser G. 
Published: July 2010

Let

f
(
z
)

f(z)

be a polynomial of degree n. Given that

f
(
z
)

f(z)

has a zero on the circle

|

z

|

=
ρ

(
0
>
ρ
>
∞
)

|z| = \rho \;(0 > \rho > \infty )

, we estimate

|

f
(
0
)

|

|f(0)|

and
\[

(

(
2
π

)

−
1

∫
0

2
π

|

f
(

e

i
θ

)

|

2

d
θ

)

1

/

2

{({(2\pi )^{ - 1}}\int _0^{2\pi }|f({e^{i\theta }}){|^2}d\theta )^{1/2}}

\]
in terms of

max

|

z

|

=
1

|

f
(
z
)

|

{\max _{|z| = 1}}|f(z)|

. We also consider some other related problems.
A special integral and a Gronwall inequality
Authors: Helton Burrell W. 
Published: July 2010

This paper considers a special integral

(
I
)

∫
a
b

(
f
d
g
+
H
)

(I)\smallint _a^b(fdg + H)

which is a subdivision-refinement-type limit of the approximating sum
\[

∑
1
n

{
f
(

t
i

)
[
g
(

x
i

)
−
g
(

x

i
−
1

)
]
+
H
(

x

i
−
1

,

x
i

)
}
,

\sum \limits _1^n {\{ f({t_i})[g({x_i}) - g({x_{i - 1}})] + H({x_{i - 1}},{x_i})\} ,}

\]
where

x

i
−
1

>

t
i

>

x
i

{x_{i - 1}} > {t_i} > {x_i}

. The author shows, with appropriate restrictions, that

(
I
)

∫
a
b

(
f
d
g
+
H
)

(I)\smallint _a^b(fdg + H)

exists if and only if
\[

(
R
)

∫
x
y

(
f
d
g
+
H
−

A
−

)
=
(
L
)

∫
x
y

(
f
d
g
+
H
+

A
+

)

(R)\smallint _x^y(fdg + H - {A^ - }) = (L)\smallint _x^y(fdg + H + {A^ + })

\]
for

a
⩽
x
>
y
⩽
b

a \leqslant x > y \leqslant b

, where

A
(
p
,
q
)
=
[
f
(
q
)
−
f
(
p
)
]
[
g
(
q
)
−
g
(
p
)
]
,

A
−

(
p
,
q
)
=
A
(

q
−

,
q
)

A(p,q) = [f(q) - f(p)][g(q) - g(p)],{A^ - }(p,q) = A({q^ - },q)

and

A
+

(
p
,
q
)
=
A
(
p
,

p
+

)

{A^ + }(p,q) = A(p,{p^ + })

. Furthermore, if either of the equivalent statements is true, then all the integrals are equal. These equivalent statements are used to prove an integration-by-parts theorem and to solve a Gronwall inequality involving this special integral. Product integrals are used in the solution of the Gronwall inequality.
Asymptotic equipartition of energy for differential equations in Hilbert space
Authors: Goldstein Jerome A. , Sandefur James T. 
Published: July 2010

Of concern are second order differential equations of the form

(
d

/

d
t
−
i

f
1

(
A
)
)
(
d

/

d
t
−
i

f
2

(
A
)
)
u
=
0

(d/dt - i{f_1}(A))(d/dt - i{f_2}(A))u = 0

. Here A is a selfadjoint operator and

f
1

,

f
2

{f_1},{f_2}

are real-valued Borel functions on the spectrum of A. The Cauchy problem for this equation is governed by a certain one parameter group of unitary operators. This group allows one to define the energy of a solution; this energy depends on the initial data but not on the time t. The energy is broken into two parts, kinetic energy

K
(
t
)

K(t)

and potential energy

P
(
t
)

P(t)

, and conditions on A,

f
1

,

f
2

{f_1},{f_2}

are given to insure asymptotic equipartition of energy:

lim

t
→
±
∞

K
(
t
)
=

lim

t
→
±
∞

P
(
t
)

{\lim _{t \to \pm \infty }}K(t) = {\lim _{t \to \pm \infty }}P(t)

for all choices of initial data. These results generalize the corresponding results of Goldstein for the abstract wave equation

d
2

u

/

d

t
2

+

A
2

u
=
0

{d^2}u/d{t^2} + {A^2}u = 0

. (In this case,

f
1

(
λ
)
≡
λ
,

f
2

(
λ
)
≡
−
λ

{f_1}(\lambda ) \equiv \lambda ,{f_2}(\lambda ) \equiv - \lambda

.)
Tauberian theorems for a positive definite form, with applications to a Volterra equation
Authors: Staffans Olof J. 
Published: July 2010

We study the relation between the condition
\[

sup

T
>
0

∫

[
0
,
T
]

φ
¯

(
t
)

∫

[
t
−
T
,
t
]

φ
(
t
−
s
)

d
v
(
s
)

d
t
>
∞

\sup \limits _{T > 0} \int _{[0,T]} {\bar \varphi (t)} \int _{[t - T,t]} {\varphi (t - s)\;dv (s)\;dt > \infty }

\]
and the asymptotic behavior of the bounded function

φ
\varphi

when

ν
\nu

is a positive definite measure. Earlier we have proved that if

ν
\nu

is strictly positive definite and

φ
\varphi

satisfies a tauberian condition, then

φ
(
t
)
→
0

\varphi (t) \to 0

as

t
→
∞

t \to \infty

. Here we characterize the spectrum of the limit set of

φ
\varphi

in the case when

ν
\nu

is not strictly positive definite. Applying this theory to a nonlinear Volterra equation we get some new results on the asymptotic behavior of its bounded solutions.
Derivatives of entire functions and a question of Pólya
Authors: Hellerstein Simon , Williamson Jack 
Published: July 2010

An old question of Pólya asks whether an entire function f which has, along with each of its derivatives, only real zeros must be of the form
\[

f
(
z
)
=

z
m

e

−
a

z
2

+
b
z
+
c

∏
n

(

1
−

z

z
n

)

e

z

/

z
n

f(z) = {z^m}{e^{ - a{z^2} + bz + c}}\prod \limits _n {\left ( {1 - \frac {z}{{{z_n}}}} \right )} {e^{z/{z_n}}}

\]
where

a
⩾
0
,
b

a \geqslant 0,b

and the

z
n

{z_n}

are real, and

Σ
n

z
n

−
2

>
∞

{\Sigma _n}z_n^{ - 2} > \infty

. This note answers this question (essentially in the affirmative) if f is of finite order; indeed, it is established that if

f
,

f
′

f,f’

, and

f
f

have only real zeros (f of finite order), then either f has the above form or f has one of the forms
\[

f
(
z
)
=
a

e

b
z

,

f
(
z
)
=
a
(

e

i
c
z

−

e

i
d

)

f(z) = a{e^{bz}},\quad f(z) = a({e^{icz}} - {e^{id}})

\]
where a, b, c, and d are constants, b complex, c and d real.
Existence of integrals and the solution of integral equations
Authors: Helton Jon C. 
Published: July 2010

Functions are from R to N or

R
×
R

R \times R

to N, where R denotes the real numbers and N denotes a normed complete ring. If S, T and G are functions from

R
×
R

R \times R

to N, each of

S
(

p
−

,
p
)
,
S
(

p
−

,

p
−

)
,
T
(

p
−

,
p
)

S({p^ - },p),S({p^ - },{p^ - }),T({p^ - },p)

and

T
(

p
−

,

p
−

)

T({p^ - },{p^ - })

exists for

a
>
p
⩽
b

a > p \leqslant b

, each of

S
(
p
,

p
+

)
,
S
(

p
+

,

p
+

)
,
T
(
p
,

p
+

)

S(p,{p^ + }),S({p^ + },{p^ + }),T(p,{p^ + })

and

T
(

p
+

,

p
+

)

T({p^ + },{p^ + })

exists for

a
⩽
p
>
b

a \leqslant p > b

, G has bounded variation on [a, b] and

∫
a
b

G

\smallint _a^bG

exists, then each of
\[

∫
a
b

S

[

G
−
∫
G

]

T

and

∫
a
b

S

[

1
+
G
−
∏

(
1
+
G
)

]

T

\int _a^b S \left [ {G - \int G } \right ]T\quad {\text {and}}\quad \int _a^b {S\left [ {1 + G - \prod {(1 + G)} } \right ]} \;T

\]
exists and is zero. These results can be used to solve integral equations without the existence of integrals of the form
\[

∫
a
b

|

G
−
∫
G

|

=
0

and

∫
a
b

|

1
+
G
−
∏

(
1
+
G
)

|

=
0.

\int _a^b {\left | {G - \int G } \right | = 0} \quad {\text {and}}\quad \int _a^b {\left | {1 + G - \prod {(1 + G)} } \right |} = 0.

\]
This is demonstrated by solving the linear integral equation
\[

f
(
x
)
=
h
(
x
)
+
(
L
R
)

∫
a
x

(
f
G
+
f
H
)

f(x) = h(x) + (LR)\int _a^x {(fG + fH)}

\]
and the Riccati integral equations
\[

f
(
x
)
=
w
(
x
)
+
(
L
R
L
R
)

∫
a
x

(
f
H
+
G
f
+
f
K
f
)

f(x) = w(x) + (LRLR)\int _a^x {(fH + Gf + fKf)}

\]
without the existence of the previously mentioned integrals.
A global theorem for singularities of maps between oriented 2-manifolds
Authors: Quine J. R. 
Published: July 2010

Let M and N be smooth compact oriented connected 2-mani-folds. Suppose

f
:
M
→
N

f:M \to N

is smooth and every point

p
∈
M

p \in M

is either a fold point, cusp point, or regular point of f i.e., f is excellent in the sense of Whitney. Let

M
+

{M^ + }

be the closure of the set of regular points at which f preserves orientation and M the closure of the set of regular points at which f reverses orientation. Let

p
1

,
…
,

p
n

{p_1}, \ldots ,{p_n}

be the cusp points and

μ
(

p
k

)

\mu ({p_k})

the local degree at the cusp point

p
k

{p_k}

. We prove the following:
\[

χ
(
M
)
−
2
χ
(

M
−

)
+
∑
μ
(

p
k

)
=
(
deg
⁡
f
)
χ
(
N
)

\chi (M) - 2\chi ({M^ - }) + \sum \mu ({p_k}) = (\deg f)\chi (N)

\]
where

χ
\chi

is the Euler characteristic and deg is the topological degree. We show that it is a generalization of the Riemann-Hurwitz formula of complex analysis and give some examples.
Some exponential moments of sums of independent random variables
Authors: Kuelbs J. 
Published: July 2010

If

{

X
n

}

\{ {X_n}\}

is a sequence of vector valued random variables,

{

a
n

}

\{ {a_n}\}

a sequence of positive constants, and

M
=

sup

n
⩾
1

‖

(

X
1

+
⋯
+

X
n

)

/

a
n

‖

M = {\sup _{n \geqslant 1}}\left \| {({X_1} + \cdots + {X_n})/{a_n}} \right \|

, we examine when

E
(
Φ
(
M
)
)
>
∞

E(\Phi (M)) > \infty

under various conditions on

Φ
,
{

X
n

}

\Phi ,\{ {X_n}\}

, and

{

a
n

}

\{ {a_n}\}

. These integrability results easily apply to empirical distribution functions.